Application of the Green's function method for 2- and 3-dimensional steady transonic flows

NASA Technical Reports Server (NTRS)

Tseng, K.

1984-01-01

A Time-Domain Green's function method for the nonlinear time-dependent three-dimensional aerodynamic potential equation is presented. The Green's theorem is being used to transform the partial differential equation into an integro-differential-delay equation. Finite-element and finite-difference methods are employed for the spatial and time discretizations to approximate the integral equation by a system of differential-delay equations. Solution may be obtained by solving for this nonlinear simultaneous system of equations in time. This paper discusses the application of the method to the Transonic Small Disturbance Equation and numerical results for lifting and nonlifting airfoils and wings in steady flows are presented.

A necessary and sufficient condition for well-posedness of initial value problems of retarded functional differential equations

NASA Astrophysics Data System (ADS)

Nishiguchi, Junya

2017-09-01

We introduce the retarded functional differential equations (RFDEs) with general delay structure to treat various delay differential equations (DDEs) in a unified way and to clarify the delay structure in those dynamics. We are interested in the question as to which space of histories is suitable for the dynamics of each DDE, and investigate the well-posedness of the initial value problems (IVPs) of the RFDEs. A main theorem is that the IVP is well-posed for any ;admissible; history functional if and only if the semigroup determined by the trivial RFDE x ˙ = 0 is continuous. We clarify the meaning of the Hale-Kato axiom (Hale & Kato [12]) by applying this result to RFDEs with infinite delay. We also apply the result to DDEs with unbounded time- and state-dependent delays.

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation.

PubMed

Ingalls, Brian; Mincheva, Maya; Roussel, Marc R

2017-07-01

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.

Estimation of delays and other parameters in nonlinear functional differential equations

NASA Technical Reports Server (NTRS)

Banks, H. T.; Lamm, P. K. D.

1983-01-01

A spline-based approximation scheme for nonlinear nonautonomous delay differential equations is discussed. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient-type parameters. A brief summary of some of the related numerical findings is also given.

Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE

NASA Astrophysics Data System (ADS)

Ansmann, Gerrit

2018-04-01

We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing the user to efficiently integrate differential equations from a higher-level interpreted language. The presented modules are particularly suited for large systems of differential equations such as those used to describe dynamics on complex networks. Through the selected method of input, the presented modules also allow almost complete automatization of the process of estimating regular as well as transversal Lyapunov exponents for ordinary and delay differential equations. We conceptually discuss the modules' design, analyze their performance, and demonstrate their capabilities by application to timely problems.

Maximum principle for a stochastic delayed system involving terminal state constraints.

PubMed

Wen, Jiaqiang; Shi, Yufeng

2017-01-01

We investigate a stochastic optimal control problem where the controlled system is depicted as a stochastic differential delayed equation; however, at the terminal time, the state is constrained in a convex set. We firstly introduce an equivalent backward delayed system depicted as a time-delayed backward stochastic differential equation. Then a stochastic maximum principle is obtained by virtue of Ekeland's variational principle. Finally, applications to a state constrained stochastic delayed linear-quadratic control model and a production-consumption choice problem are studied to illustrate the main obtained result.

Stability analysis for a delay differential equations model of a hydraulic turbine speed governor

NASA Astrophysics Data System (ADS)

Halanay, Andrei; Safta, Carmen A.; Dragoi, Constantin; Piraianu, Vlad F.

2017-01-01

The paper aims to study the dynamic behavior of a speed governor for a hydraulic turbine using a mathematical model. The nonlinear mathematical model proposed consists in a system of delay differential equations (DDE) to be compared with already established mathematical models of ordinary differential equations (ODE). A new kind of nonlinearity is introduced as a time delay. The delays can characterize different running conditions of the speed governor. For example, it is considered that spool displacement of hydraulic amplifier might be blocked due to oil impurities in the oil supply system and so the hydraulic amplifier has a time delay in comparison to the time control. Numerical simulations are presented in a comparative manner. A stability analysis of the hydraulic control system is performed, too. Conclusions of the dynamic behavior using the DDE model of a hydraulic turbine speed governor are useful in modeling and controlling hydropower plants.

Local bifurcations in differential equations with state-dependent delay.

PubMed

Sieber, Jan

2017-11-01

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.

Stability and bifurcation analysis of a generalized scalar delay differential equation.

PubMed

Bhalekar, Sachin

2016-08-01

This paper deals with the stability and bifurcation analysis of a general form of equation D(α)x(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.

Computational Algorithms or Identification of Distributed Parameter Systems

DTIC Science & Technology

1993-04-24

delay-differential equations, Volterra integral equations, and partial differential equations with memory terms . In particular we investigated a...tested for estimating parameters in a Volterra integral equation arising from a viscoelastic model of a flexible structure with Boltzmann damping. In...particular, one of the parameters identified was the order of the derivative in Volterra integro-differential equations containing fractional

Local and global Hopf bifurcation analysis in a neutral-type neuron system with two delays

NASA Astrophysics Data System (ADS)

Lv, Qiuyu; Liao, Xiaofeng

2018-03-01

In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting τ1, τ2 respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.

Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay

NASA Astrophysics Data System (ADS)

Tang, Xianhua; Cao, Daomin; Zou, Xingfu

We consider a periodic Lotka-Volterra competition system without instantaneous negative feedbacks (i.e., pure-delay systems) x(t)=x(t)[r(t)-∑j=1na(t)x(t-τ(t))], i=1,2,…,n. We establish some 3/2-type criteria for global attractivity of a positive periodic solution of the system, which generalize the well-known Wright's 3/2 criteria for the autonomous delay logistic equation, and thereby, address the open problem proposed by both Kuang [Y. Kuang, Global stability in delayed nonautonomous Lotka-Volterra type systems without saturated equilibria, Differential Integral Equations 9 (1996) 557-567] and Teng [Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Differential Equations 179 (2002) 538-561].

Numerical Bifurcation Analysis of Delayed Recycle Stream in a Continuously Stirred Tank Reactor

NASA Astrophysics Data System (ADS)

Gangadhar, Nalwala Rohitbabu; Balasubramanian, Periyasamy

2010-10-01

In this paper, we present the stability analysis of delay differential equations which arise as a result of transportation lag in the CSTR-mechanical separator recycle system. A first order irreversible elementary reaction is considered to model the system and is governed by the delay differential equations. The DDE-BIFTOOL software package is used to analyze the stability of the delay system. The present analysis reveals that the system exhibits delay independent stability for isothermal operation of the CSTR. In the absence of delay, the system is dynamically unstable for non-isothermal operation of the CSTR, and as a result of delay, the system exhibits delay dependent stability.

Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter.

PubMed

Yi, Sun; Nelson, Patrick W; Ulsoy, A Galip

2007-04-01

In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to be useful for solving scalar first-order DDEs, has recently been extended to a matrix Lambert W function approach to solve systems of DDEs. The essential advantages of the matrix Lambert W approach are not only the similarity to the concept of the state transition matrix in lin ear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay, which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert W function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Factorization and the synthesis of optimal feedback kernels for differential-delay systems

NASA Technical Reports Server (NTRS)

Milman, Mark M.; Scheid, Robert E.

1987-01-01

A combination of ideas from the theories of operator Riccati equations and Volterra factorizations leads to the derivation of a novel, relatively simple set of hyperbolic equations which characterize the optimal feedback kernel for the finite-time regulator problem for autonomous differential-delay systems. Analysis of these equations elucidates the underlying structure of the feedback kernel and leads to the development of fast and accurate numerical methods for its computation. Unlike traditional formulations based on the operator Riccati equation, the gain is characterized by means of classical solutions of the derived set of equations. This leads to the development of approximation schemes which are analogous to what has been accomplished for systems of ordinary differential equations with given initial conditions.

A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations.

PubMed

Benhammouda, Brahim; Vazquez-Leal, Hector

2016-01-01

This work presents an analytical solution of some nonlinear delay differential equations (DDEs) with variable delays. Such DDEs are difficult to treat numerically and cannot be solved by existing general purpose codes. A new method of steps combined with the differential transform method (DTM) is proposed as a powerful tool to solve these DDEs. This method reduces the DDEs to ordinary differential equations that are then solved by the DTM. Furthermore, we show that the solutions can be improved by Laplace-Padé resummation method. Two examples are presented to show the efficiency of the proposed technique. The main advantage of this technique is that it possesses a simple procedure based on a few straight forward steps and can be combined with any analytical method, other than the DTM, like the homotopy perturbation method.

New stability conditions for mixed linear Levin-Nohel integro-differential equations

NASA Astrophysics Data System (ADS)

Dung, Nguyen Tien

2013-08-01

For the mixed Levin-Nohel integro-differential equation, we obtain new necessary and sufficient conditions of asymptotic stability. These results improve those obtained by Becker and Burton ["Stability, fixed points and inverse of delays," Proc. - R. Soc. Edinburgh, Sect. A 136, 245-275 (2006)], 10.1017/S0308210500004546 and Jin and Luo ["Stability of an integro-differential equation," Comput. Math. Appl. 57(7), 1080-1088 (2009)], 10.1016/j.camwa.2009.01.006 when b(t) = 0 and supplement the 3/2-stability theorem when a(t, s) = 0. In addition, the case of the equations with several delays is discussed as well.

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay.

PubMed

Korkmaz, Erdal

2017-01-01

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature.

Constructing Hopf bifurcation lines for the stability of nonlinear systems with two time delays

NASA Astrophysics Data System (ADS)

Nguimdo, Romain Modeste

2018-03-01

Although the plethora real-life systems modeled by nonlinear systems with two independent time delays, the algebraic expressions for determining the stability of their fixed points remain the Achilles' heel. Typically, the approach for studying the stability of delay systems consists in finding the bifurcation lines separating the stable and unstable parameter regions. This work deals with the parametric construction of algebraic expressions and their use for the determination of the stability boundaries of fixed points in nonlinear systems with two independent time delays. In particular, we concentrate on the cases for which the stability of the fixed points can be ascertained from a characteristic equation corresponding to that of scalar two-delay differential equations, one-component dual-delay feedback, or nonscalar differential equations with two delays for which the characteristic equation for the stability analysis can be reduced to that of a scalar case. Then, we apply our obtained algebraic expressions to identify either the parameter regions of stable microwaves generated by dual-delay optoelectronic oscillators or the regions of amplitude death in identical coupled oscillators.

Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gupta, Chinmaya; López, José Manuel; Azencott, Robert

Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemicalmore » Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.« less

Model Predictive Optimal Control of a Time-Delay Distributed-Parameter Systems

NASA Technical Reports Server (NTRS)

Nguyen, Nhan

2006-01-01

This paper presents an optimal control method for a class of distributed-parameter systems governed by first order, quasilinear hyperbolic partial differential equations that arise in many physical systems. Such systems are characterized by time delays since information is transported from one state to another by wave propagation. A general closed-loop hyperbolic transport model is controlled by a boundary control embedded in a periodic boundary condition. The boundary control is subject to a nonlinear differential equation constraint that models actuator dynamics of the system. The hyperbolic equation is thus coupled with the ordinary differential equation via the boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to implement a model predictive control design for a wind tunnel to eliminate a transport delay effect that causes a poor Mach number regulation.

Bifurcation to large period oscillations in physical systems controlled by delay

NASA Astrophysics Data System (ADS)

Erneux, Thomas; Walther, Hans-Otto

2005-12-01

An unusual bifurcation to time-periodic oscillations of a class of delay differential equations is investigated. As we approach the bifurcation point, both the amplitude and the frequency of the oscillations go to zero. The class of delay differential equations is a nonlinear extension of a nonevasive control method and is motivated by a recent study of the foreign exchange rate oscillations. By using asymptotic methods, we determine the bifurcation scaling laws for the amplitude and the period of the oscillations.

On exponential stability of linear Levin-Nohel integro-differential equations

NASA Astrophysics Data System (ADS)

Tien Dung, Nguyen

2015-02-01

The aim of this paper is to investigate the exponential stability for linear Levin-Nohel integro-differential equations with time-varying delays. To the best of our knowledge, the exponential stability for such equations has not yet been discussed. In addition, since we do not require that the kernel and delay are continuous, our results improve those obtained in Becker and Burton [Proc. R. Soc. Edinburgh, Sect. A: Math. 136, 245-275 (2006)]; Dung [J. Math. Phys. 54, 082705 (2013)]; and Jin and Luo [Comput. Math. Appl. 57(7), 1080-1088 (2009)].

Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations.

PubMed

Banks, H Thomas; Robbins, Danielle; Sutton, Karyn L

2013-01-01

In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.

The application of Legendre-tau approximation to parameter identification for delay and partial differential equations

NASA Technical Reports Server (NTRS)

Ito, K.

1983-01-01

Approximation schemes based on Legendre-tau approximation are developed for application to parameter identification problem for delay and partial differential equations. The tau method is based on representing the approximate solution as a truncated series of orthonormal functions. The characteristic feature of the Legendre-tau approach is that when the solution to a problem is infinitely differentiable, the rate of convergence is faster than any finite power of 1/N; higher accuracy is thus achieved, making the approach suitable for small N.

A differential delay equation arising from the sieve of Eratosthenes

NASA Technical Reports Server (NTRS)

Cheer, A. Y.; Goldston, D. A.

1990-01-01

Consideration is given to the differential delay equation introduced by Buchstab (1937) in connection with an asymptotic formula for the uncanceled terms in the sieve of Eratosthenes. Maier (1985) used this result to show there is unexpected irreqularity in the distribution of primes in short intervals. The function omega(u) is studied in this paper using numerical and analytical techniques. The results are applied to give some numerical constants in Maier's theorem.

Numerical modelling in biosciences using delay differential equations

NASA Astrophysics Data System (ADS)

Bocharov, Gennadii A.; Rihan, Fathalla A.

2000-12-01

Our principal purposes here are (i) to consider, from the perspective of applied mathematics, models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics, (ii) to review the application of numerical techniques to investigate these models. We show that there are prima facie reasons for using such models: (i) they have a richer mathematical framework (compared with ordinary differential equations) for the analysis of biosystem dynamics, (ii) they display better consistency with the nature of certain biological processes and predictive results. We analyze both the qualitative and quantitative role that delays play in basic time-lag models proposed in population dynamics, epidemiology, physiology, immunology, neural networks and cell kinetics. We then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences, comparing them with those implemented by the bio-modellers.

Alternans promotion in cardiac electrophysiology models by delay differential equations.

PubMed

Gomes, Johnny M; Dos Santos, Rodrigo Weber; Cherry, Elizabeth M

2017-09-01

Cardiac electrical alternans is a state of alternation between long and short action potentials and is frequently associated with harmful cardiac conditions. Different dynamic mechanisms can give rise to alternans; however, many cardiac models based on ordinary differential equations are not able to reproduce this phenomenon. A previous study showed that alternans can be induced by the introduction of delay differential equations (DDEs) in the formulations of the ion channel gating variables of a canine myocyte model. The present work demonstrates that this technique is not model-specific by successfully promoting alternans using DDEs for five cardiac electrophysiology models that describe different types of myocytes, with varying degrees of complexity. By analyzing results across the different models, we observe two potential requirements for alternans promotion via DDEs for ionic gates: (i) the gate must have a significant influence on the action potential duration and (ii) a delay must significantly impair the gate's recovery between consecutive action potentials.

Sensitivity analysis of dynamic biological systems with time-delays.

PubMed

Wu, Wu Hsiung; Wang, Feng Sheng; Chang, Maw Shang

2010-10-15

Mathematical modeling has been applied to the study and analysis of complex biological systems for a long time. Some processes in biological systems, such as the gene expression and feedback control in signal transduction networks, involve a time delay. These systems are represented as delay differential equation (DDE) models. Numerical sensitivity analysis of a DDE model by the direct method requires the solutions of model and sensitivity equations with time-delays. The major effort is the computation of Jacobian matrix when computing the solution of sensitivity equations. The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors. To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary. We have proposed an efficient algorithm with an adaptive step size control to compute the solution and dynamic sensitivities of biological systems described by ordinal differential equations (ODEs). The adaptive direct-decoupled algorithm is extended to solve the solution and dynamic sensitivities of time-delay systems describing by DDEs. To save the human effort and avoid the human errors in the computation of partial derivatives, an automatic differentiation technique is embedded in the extended algorithm to evaluate the Jacobian matrix. The extended algorithm is implemented and applied to two realistic models with time-delays: the cardiovascular control system and the TNF-α signal transduction network. The results show that the extended algorithm is a good tool for dynamic sensitivity analysis on DDE models with less user intervention. By comparing with direct-coupled methods in theory, the extended algorithm is efficient, accurate, and easy to use for end users without programming background to do dynamic sensitivity analysis on complex biological systems with time-delays.

Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition.

PubMed

Vlad, Marcel Ovidiu; Ross, John

2002-12-01

We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

PubMed

Khader, M M

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bi, Ping; Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Rd., Shanghai 200241; Ruan, Shigui, E-mail: ruan@math.miami.edu

2014-06-15

In this paper, a tumor and immune system interaction model consisted of two differential equations with three time delays is considered in which the delays describe the proliferation of tumor cells, the process of effector cells growth stimulated by tumor cells, and the differentiation of immune effector cells, respectively. Conditions for the asymptotic stability of equilibria and existence of Hopf bifurcations are obtained by analyzing the roots of a second degree exponential polynomial characteristic equation with delay dependent coefficients. It is shown that the positive equilibrium is asymptotically stable if all three delays are less than their corresponding critical valuesmore » and Hopf bifurcations occur if any one of these delays passes through its critical value. Numerical simulations are carried out to illustrate the rich dynamical behavior of the model with different delay values including the existence of regular and irregular long periodic oscillations.« less

Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales

NASA Astrophysics Data System (ADS)

Han, Zhenlai; Sun, Shurong; Shi, Bao

2007-10-01

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden-Fowler delay dynamic equationsx[Delta][Delta](t)+p(t)x[gamma]([tau](t))=0 on a time scale ; here [gamma] is a quotient of odd positive integers with p(t) real-valued positive rd-continuous functions defined on . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales. Our results in this paper not only extend the results given in [R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second-order delay dynamic equations, Can. Appl. Math. Q. 13 (1) (2005) 1-18] but also unify the oscillation of the second-order Emden-Fowler delay differential equation and the second-order Emden-Fowler delay difference equation.

Dynamics of a neutral delay equation for an insect population with long larval and short adult phases

NASA Astrophysics Data System (ADS)

Gourley, Stephen A.; Kuang, Yang

We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.

Response of an oscillatory differential delay equation to a single stimulus.

PubMed

Mackey, Michael C; Tyran-Kamińska, Marta; Walther, Hans-Otto

2017-04-01

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

Uncertainty in Damage Detection, Dynamic Propagation and Just-in-Time Networks

DTIC Science & Technology

2015-08-03

estimated parameter uncertainty in dynamic data sets; high order compact finite difference schemes for Helmholtz equations with discontinuous wave numbers...delay differential equations with a Gamma distributed delay. We found that with the same population size the histogram plots for the solution to the...schemes for Helmholtz equations with discontinuous wave numbers across interfaces. • We carried out numerical sensitivity analysis with respect to

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

DOE Office of Scientific and Technical Information (OSTI.GOV)

Boubendir, Yassine; Mendez, Vicenc; Rotstein, Horacio G.

2010-09-15

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delaymore » time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.« less

Estimation of coupling between time-delay systems from time series

NASA Astrophysics Data System (ADS)

Prokhorov, M. D.; Ponomarenko, V. I.

2005-07-01

We propose a method for estimation of coupling between the systems governed by scalar time-delay differential equations of the Mackey-Glass type from the observed time series data. The method allows one to detect the presence of certain types of linear coupling between two time-delay systems, to define the type, strength, and direction of coupling, and to recover the model equations of coupled time-delay systems from chaotic time series corrupted by noise. We verify our method using both numerical and experimental data.

Stability Criteria for Differential Equations with Variable Time Delays

ERIC Educational Resources Information Center

Schley, D.; Shail, R.; Gourley, S. A.

2002-01-01

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay…

Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain

NASA Astrophysics Data System (ADS)

Yi, Taishan; Zou, Xingfu

In this paper, we study the global dynamics of a class of differential equations with temporal delay and spatial non-locality in an unbounded domain. Adopting the compact open topology, we describe the delicate asymptotic properties of the nonlocal delayed effect and establish some a priori estimate for nontrivial solutions which enables us to show the permanence of the equation. Combining these results with a dynamical systems approach, we determine the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker's birth function and Mackey-Glass's hematopoiesis function, we obtain threshold results for the global dynamics of these two models. We explain why our results on the global attractivity of the positive equilibrium in C∖{0} under the compact open topology becomes invalid in C∖{0} with respect to the usual supremum norm, and we identify a subset of C∖{0} in which the positive equilibrium remains attractive with respect to the supremum norm.

Ultimate boundedness stability and controllability of hereditary systems

NASA Technical Reports Server (NTRS)

Chukwu, E. N.

1979-01-01

By generalizing the Liapunov-Yoshizawa techniques, necessary and sufficient conditions are given for uniform boundedness and uniform ultimate boundedness of a rather general class of nonlinear differential equations of neutral type. Among the applications treated by the methods are the Lienard equation of neutral type and hereditary systems of Lurie type. The absolute stability of this later equation is also investigated. A certain existence result of a solution of a neutral functional differential inclusion with two point boundary values is applied to study the exact function space controllability of a nonlinear neutral functional differential control system. A geometric growth condition is used to characterize both the function space and Euclidean controllability of another nonlinear delay system which has a compact and convex control set. This yields conditions under which perturbed nonlinear delay controllable systems are controllable.

Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

NASA Astrophysics Data System (ADS)

Song, Yongli; Zhang, Tonghua; Tadé, Moses O.

2009-12-01

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.

Transit and lifespan in neutrophil production: implications for drug intervention.

PubMed

Câmara De Souza, Daniel; Craig, Morgan; Cassidy, Tyler; Li, Jun; Nekka, Fahima; Bélair, Jacques; Humphries, Antony R

2018-02-01

A comparison of the transit compartment ordinary differential equation modelling approach to distributed and discrete delay differential equation models is studied by focusing on Quartino's extension to the Friberg transit compartment model of myelosuppression, widely relied upon in the pharmaceutical sciences to predict the neutrophil response after chemotherapy, and on a QSP delay differential equation model of granulopoiesis. An extension to the Quartino model is provided by considering a general number of transit compartments and introducing an extra parameter that allows for the decoupling of the maturation time from the production rate of cells. An overview of the well established linear chain technique, used to reformulate transit compartment models with constant transit rates as distributed delay differential equations (DDEs), is then given. A state-dependent time rescaling of the Quartino model is performed to apply the linear chain technique and rewrite the Quartino model as a distributed DDE, yielding a discrete DDE model in a certain parameter limit. Next, stability and bifurcation analyses are undertaken in an effort to situate such studies in a mathematical pharmacology context. We show that both the original Friberg and the Quartino extension models incorrectly define the mean maturation time, essentially treating the proliferative pool as an additional maturation compartment. This misspecification can have far reaching consequences on the development of future models of myelosuppression in PK/PD.

A differential delay equation arising from the sieve of Eratosthenes

NASA Astrophysics Data System (ADS)

Cheer, A. Y.; Goldston, D. A.

1990-07-01

The differential delay equation defined by ω (u) = 1/u for 1 ≤ u ≤ 2 and (uω (u))' = ω (u - 1) for u ≥ 2 was introduced by Buchstab in connection with an asymptotic formula for the number of uncanceled terms in the sieve of Eratosthenes. Maier has recently used this result to show there is unexpected irregularity in the distribution of primes in short intervals. The function ω (u) is studied in this paper using numerical and analytical techniques. The results are applied to give some numerical constants in Maier's theorem.

2–stage stochastic Runge–Kutta for stochastic delay differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rosli, Norhayati; Jusoh Awang, Rahimah; Bahar, Arifah

2015-05-15

This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.

Causal structure of oscillations in gene regulatory networks: Boolean analysis of ordinary differential equation attractors.

PubMed

Sun, Mengyang; Cheng, Xianrui; Socolar, Joshua E S

2013-06-01

A common approach to the modeling of gene regulatory networks is to represent activating or repressing interactions using ordinary differential equations for target gene concentrations that include Hill function dependences on regulator gene concentrations. An alternative formulation represents the same interactions using Boolean logic with time delays associated with each network link. We consider the attractors that emerge from the two types of models in the case of a simple but nontrivial network: a figure-8 network with one positive and one negative feedback loop. We show that the different modeling approaches give rise to the same qualitative set of attractors with the exception of a possible fixed point in the ordinary differential equation model in which concentrations sit at intermediate values. The properties of the attractors are most easily understood from the Boolean perspective, suggesting that time-delay Boolean modeling is a useful tool for understanding the logic of regulatory networks.

Maximum profile likelihood estimation of differential equation parameters through model based smoothing state estimates.

PubMed

Campbell, D A; Chkrebtii, O

2013-12-01

Statistical inference for biochemical models often faces a variety of characteristic challenges. In this paper we examine state and parameter estimation for the JAK-STAT intracellular signalling mechanism, which exemplifies the implementation intricacies common in many biochemical inference problems. We introduce an extension to the Generalized Smoothing approach for estimating delay differential equation models, addressing selection of complexity parameters, choice of the basis system, and appropriate optimization strategies. Motivated by the JAK-STAT system, we further extend the generalized smoothing approach to consider a nonlinear observation process with additional unknown parameters, and highlight how the approach handles unobserved states and unevenly spaced observations. The methodology developed is generally applicable to problems of estimation for differential equation models with delays, unobserved states, nonlinear observation processes, and partially observed histories. Crown Copyright © 2013. Published by Elsevier Inc. All rights reserved.

Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics, production version (SOUSSA-P 1.1). Volume 1: Theoretical manual. [Green function

NASA Technical Reports Server (NTRS)

Morino, L.

1980-01-01

Recent developments of the Green's function method and the computer program SOUSSA (Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics) are reviewed and summarized. Applying the Green's function method to the fully unsteady (transient) potential equation yields an integro-differential-delay equation. With spatial discretization by the finite-element method, this equation is approximated by a set of differential-delay equations in time. Time solution by Laplace transform yields a matrix relating the velocity potential to the normal wash. Premultiplying and postmultiplying by the matrices relating generalized forces to the potential and the normal wash to the generalized coordinates one obtains the matrix of the generalized aerodynamic forces. The frequency and mode-shape dependence of this matrix makes the program SOUSSA useful for multiple frequency and repeated mode-shape evaluations.

Comparative analysis of Goodwin's business cycle models

NASA Astrophysics Data System (ADS)

Antonova, A. O.; Reznik, S.; Todorov, M. D.

2016-10-01

We compare the behavior of solutions of Goodwin's business cycle equation in the form of neutral delay differential equation with fixed delay (NDDE model) and in the form of the differential equations of 3rd, 4th and 5th orders (ODE model's). Such ODE model's (Taylor series expansion of NDDE in powers of θ) are proposed in N. Dharmaraj and K. Vela Velupillai [6] for investigation of the short periodic sawthooth oscillations in NDDE. We show that the ODE's of 3rd, 4th and 5th order may approximate the asymptotic behavior of only main Goodwin's mode, but not the sawthooth modes. If the order of the Taylor series expansion exceeds 5, then the approximate ODE becomes unstable independently of time lag θ.

Spline approximations for nonlinear hereditary control systems

NASA Technical Reports Server (NTRS)

Daniel, P. L.

1982-01-01

A sline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.

Reconstruction of ensembles of coupled time-delay systems from time series.

PubMed

Sysoev, I V; Prokhorov, M D; Ponomarenko, V I; Bezruchko, B P

2014-06-01

We propose a method to recover from time series the parameters of coupled time-delay systems and the architecture of couplings between them. The method is based on a reconstruction of model delay-differential equations and estimation of statistical significance of couplings. It can be applied to networks composed of nonidentical nodes with an arbitrary number of unidirectional and bidirectional couplings. We test our method on chaotic and periodic time series produced by model equations of ensembles of diffusively coupled time-delay systems in the presence of noise, and apply it to experimental time series obtained from electronic oscillators with delayed feedback coupled by resistors.

Periodicity computation of generalized mathematical biology problems involving delay differential equations.

PubMed

Jasim Mohammed, M; Ibrahim, Rabha W; Ahmad, M Z

2017-03-01

In this paper, we consider a low initial population model. Our aim is to study the periodicity computation of this model by using neutral differential equations, which are recognized in various studies including biology. We generalize the neutral Rayleigh equation for the third-order by exploiting the model of fractional calculus, in particular the Riemann-Liouville differential operator. We establish the existence and uniqueness of a periodic computational outcome. The technique depends on the continuation theorem of the coincidence degree theory. Besides, an example is presented to demonstrate the finding.

A Factorization Approach to the Linear Regulator Quadratic Cost Problem

NASA Technical Reports Server (NTRS)

Milman, M. H.

1985-01-01

A factorization approach to the linear regulator quadratic cost problem is developed. This approach makes some new connections between optimal control, factorization, Riccati equations and certain Wiener-Hopf operator equations. Applications of the theory to systems describable by evolution equations in Hilbert space and differential delay equations in Euclidean space are presented.

Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback

NASA Astrophysics Data System (ADS)

Gurevich, Svetlana V.

2013-05-01

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Approximate controllability of a system of parabolic equations with delay

NASA Astrophysics Data System (ADS)

Carrasco, Alexander; Leiva, Hugo

2008-09-01

In this paper we give necessary and sufficient conditions for the approximate controllability of the following system of parabolic equations with delay: where [Omega] is a bounded domain in , D is an n×n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the control and B[set membership, variant]L(U,Z) with , . The standard notation zt(x) defines a function from [-[tau],0] to (with x fixed) by zt(x)(s)=z(t+s,x), -[tau][less-than-or-equals, slant]s[less-than-or-equals, slant]0. Here [tau][greater-or-equal, slanted]0 is the maximum delay, which is supposed to be finite. We assume that the operator is linear and bounded, and [phi]0[set membership, variant]Z, [phi][set membership, variant]L2([-[tau],0];Z). To this end: First, we reformulate this system into a standard first-order delay equation. Secondly, the semigroup associated with the first-order delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the Laplacian operator (); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the approximate controllability of delay system to derive our main result.

Theory of repetitively pulsed operation of diode lasers subject to delayed feedback

DOE Office of Scientific and Technical Information (OSTI.GOV)

Napartovich, A P; Sukharev, A G

2015-03-31

Repetitively pulsed operation of a diode laser with delayed feedback has been studied theoretically at varying feedback parameters and pump power levels. A new approach has been proposed that allows one to reduce the system of Lang–Kobayashi equations for a steady-state repetitively pulsed operation mode to a first-order nonlinear differential equation. We present partial solutions that allow the pulse shape to be predicted. (lasers)

Optimal control strategy for an impulsive stochastic competition system with time delays and jumps

NASA Astrophysics Data System (ADS)

Liu, Lidan; Meng, Xinzhu; Zhang, Tonghua

2017-07-01

Driven by both white and jump noises, a stochastic delayed model with two competitive species in a polluted environment is proposed and investigated. By using the comparison theorem of stochastic differential equations and limit superior theory, sufficient conditions for persistence in mean and extinction of two species are established. In addition, we obtain that the system is asymptotically stable in distribution by using ergodic method. Furthermore, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are derived from Hessian matrix method and optimal harvesting theory of differential equations. Finally, some numerical simulations are provided to illustrate the theoretical results.

Bistability and State Transition of a Delay Differential Equation Model of Neutrophil Dynamics

NASA Astrophysics Data System (ADS)

Ma, Suqi; Zhu, Kaiyi; Lei, Jinzhi

This paper studies the existence of bistable states and control strategies to induce state transitions of a delay differential equation model of neutrophil dynamics. We seek the conditions that a stable steady state and an oscillatory state coexist in the neutrophil dynamical system. Physiologically, stable steady state represents the healthy state, while oscillatory state is usually associated with diseases such as cyclical neutropenia. We study the control strategies to induce the transitions from the disease state to the healthy state by introducing temporal perturbations to system parameters. This study is valuable in designing clinical protocols for the treatment of cyclical neutropenia.

Delay chemical master equation: direct and closed-form solutions

PubMed Central

Leier, Andre; Marquez-Lago, Tatiana T.

2015-01-01

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived. PMID:26345616

Delay chemical master equation: direct and closed-form solutions.

PubMed

Leier, Andre; Marquez-Lago, Tatiana T

2015-07-08

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

Solving delay differential equations in S-ADAPT by method of steps.

PubMed

Bauer, Robert J; Mo, Gary; Krzyzanski, Wojciech

2013-09-01

S-ADAPT is a version of the ADAPT program that contains additional simulation and optimization abilities such as parametric population analysis. S-ADAPT utilizes LSODA to solve ordinary differential equations (ODEs), an algorithm designed for large dimension non-stiff and stiff problems. However, S-ADAPT does not have a solver for delay differential equations (DDEs). Our objective was to implement in S-ADAPT a DDE solver using the methods of steps. The method of steps allows one to solve virtually any DDE system by transforming it to an ODE system. The solver was validated for scalar linear DDEs with one delay and bolus and infusion inputs for which explicit analytic solutions were derived. Solutions of nonlinear DDE problems coded in S-ADAPT were validated by comparing them with ones obtained by the MATLAB DDE solver dde23. The estimation of parameters was tested on the MATLB simulated population pharmacodynamics data. The comparison of S-ADAPT generated solutions for DDE problems with the explicit solutions as well as MATLAB produced solutions which agreed to at least 7 significant digits. The population parameter estimates from using importance sampling expectation-maximization in S-ADAPT agreed with ones used to generate the data. Published by Elsevier Ireland Ltd.

Unraveling mirror properties in time-delayed quantum feedback scenarios

NASA Astrophysics Data System (ADS)

Faulstich, Fabian M.; Kraft, Manuel; Carmele, Alexander

2018-06-01

We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field and a single two-level system in front of the dielectric. The dielectric is modelled as a system of identical two-state atoms. The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf-Wigner limit. Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. Our derivation motivates and justifies the typical phenomenologicalassumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics.

Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves

NASA Astrophysics Data System (ADS)

Yi, Taishan; Chen, Yuming; Wu, Jianhong

Reaction diffusion equations with delayed nonlinear reaction terms are used as prototypes to motivate an appropriate abstract formulation of dynamical systems with unimodal nonlinearity. For such non-monotone dynamical systems, we develop a general comparison principle and show how this general comparison principle, coupled with some existing results for monotone dynamical systems, can be used to establish results on the asymptotic speeds of spread and travelling waves. We illustrate our main results by an integral equation which includes a nonlocal delayed reaction diffusion equation and a nonlocal delayed lattice differential system in an unbounded domain, with the non-monotone nonlinearities including the Ricker birth function and the Mackey-Glass hematopoiesis feedback.

Hybrid control of the Neimark-Sacker bifurcation in a delayed Nicholson's blowflies equation.

PubMed

Wang, Yuanyuan; Wang, Lisha

In this article, for delayed Nicholson's blowflies equation, we propose a hybrid control nonstandard finite-difference (NSFD) scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. Firstly, the local stability of the positive equilibria for hybrid control delay differential equation is discussed according to Hopf bifurcation theory. Then, for any step-size, a hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired point. Finally, numerical simulation results confirm that the control strategy is efficient in controlling the Neimark-Sacker bifurcation. At the same time, the results show that the NSFD control scheme is better than the Euler control method.

Two Different Approaches to Nonzero-Sum Stochastic Differential Games

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rainer, Catherine

2007-06-15

We make the link between two approaches to Nash equilibria for nonzero-sum stochastic differential games: the first one using backward stochastic differential equations and the second one using strategies with delay. We prove that, when both exist, the two notions of Nash equilibria coincide.

Non-autonomous equations with unpredictable solutions

NASA Astrophysics Data System (ADS)

Akhmet, Marat; Fen, Mehmet Onur

2018-06-01

To make research of chaos more amenable to investigating differential and discrete equations, we introduce the concepts of an unpredictable function and sequence. The topology of uniform convergence on compact sets is applied to define unpredictable functions [1,2]. The unpredictable sequence is defined as a specific unpredictable function on the set of integers. The definitions are convenient to be verified as solutions of differential and discrete equations. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a delay differential equation are proved as well as for quasilinear discrete systems. As a corollary of the theorem, a similar assertion for a quasilinear ordinary differential equation is formulated. The results are demonstrated numerically, and an application to Hopfield neural networks is provided. In particular, Poincaré chaos near periodic orbits is observed. The completed research contributes to the theory of chaos as well as to the theory of differential and discrete equations, considering unpredictable solutions.

Stochastic parameter estimation in nonlinear time-delayed vibratory systems with distributed delay

NASA Astrophysics Data System (ADS)

Torkamani, Shahab; Butcher, Eric A.

2013-07-01

The stochastic estimation of parameters and states in linear and nonlinear time-delayed vibratory systems with distributed delay is explored. The approach consists of first employing a continuous time approximation to approximate the delayed integro-differential system with a large set of ordinary differential equations having stochastic excitations. Then the problem of state and parameter estimation in the resulting stochastic ordinary differential system is represented as an optimal filtering problem using a state augmentation technique. By adapting the extended Kalman-Bucy filter to the augmented filtering problem, the unknown parameters of the time-delayed system are estimated from noise-corrupted, possibly incomplete measurements of the states. Similarly, the upper bound of the distributed delay can also be estimated by the proposed technique. As an illustrative example to a practical problem in vibrations, the parameter, delay upper bound, and state estimation from noise-corrupted measurements in a distributed force model widely used for modeling machine tool vibrations in the turning operation is investigated.

Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators

NASA Astrophysics Data System (ADS)

Meng, Xin-You; Huo, Hai-Feng; Zhang, Xiao-Bing

2011-11-01

This paper is concerned with a predator-prey system with Holling II functional response and hunting delay and gestation. By regarding the sum of delays as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. We obtained explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation. Using a global Hopf bifurcation result of Wu [Wu JH. Symmetric functional differential equations and neural networks with memory, Trans Amer Math Soc 1998;350:4799-4838] for functional differential equations, we may show the global existence of the periodic solutions. Finally, several numerical simulations illustrating the theoretical analysis are also given.

Global exponential synchronization of inertial memristive neural networks with time-varying delay via nonlinear controller.

PubMed

Gong, Shuqing; Yang, Shaofu; Guo, Zhenyuan; Huang, Tingwen

2018-06-01

The paper is concerned with the synchronization problem of inertial memristive neural networks with time-varying delay. First, by choosing a proper variable substitution, inertial memristive neural networks described by second-order differential equations can be transformed into first-order differential equations. Then, a novel controller with a linear diffusive term and discontinuous sign term is designed. By using the controller, the sufficient conditions for assuring the global exponential synchronization of the derive and response neural networks are derived based on Lyapunov stability theory and some inequality techniques. Finally, several numerical simulations are provided to substantiate the effectiveness of the theoretical results. Copyright © 2018 Elsevier Ltd. All rights reserved.

A feedback control model for network flow with multiple pure time delays

NASA Technical Reports Server (NTRS)

Press, J.

1972-01-01

A control model describing a network flow hindered by multiple pure time (or transport) delays is formulated. Feedbacks connect each desired output with a single control sector situated at the origin. The dynamic formulation invokes the use of differential difference equations. This causes the characteristic equation of the model to consist of transcendental functions instead of a common algebraic polynomial. A general graphical criterion is developed to evaluate the stability of such a problem. A digital computer simulation confirms the validity of such criterion. An optimal decision making process with multiple delays is presented.

A partial differential equation model and its reduction to an ordinary differential equation model for prostate tumor growth under intermittent hormone therapy.

PubMed

Tao, Youshan; Guo, Qian; Aihara, Kazuyuki

2014-10-01

Hormonal therapy with androgen suppression is a common treatment for advanced prostate tumors. The emergence of androgen-independent cells, however, leads to a tumor relapse under a condition of long-term androgen deprivation. Clinical trials suggest that intermittent androgen suppression (IAS) with alternating on- and off-treatment periods can delay the relapse when compared with continuous androgen suppression (CAS). In this paper, we propose a mathematical model for prostate tumor growth under IAS therapy. The model elucidates initial hormone sensitivity, an eventual relapse of a tumor under CAS therapy, and a delay of a relapse under IAS therapy, which are due to the coexistence of androgen-dependent cells, androgen-independent cells resulting from reversible changes by adaptation, and androgen-independent cells resulting from irreversible changes by genetic mutations. The model is formulated as a free boundary problem of partial differential equations that describe the evolution of populations of the abovementioned three types of cells during on-treatment periods and off-treatment periods. Moreover, the model can be transformed into a piecewise linear ordinary differential equation model by introducing three new volume variables, and the study of the resulting model may help to devise optimal IAS schedules.

A model of a fishery with fish stock involving delay equations.

PubMed

Auger, P; Ducrot, Arnaud

2009-12-13

The aim of this paper is to provide a new mathematical model for a fishery by including a stock variable for the resource. This model takes the form of an infinite delay differential equation. It is mathematically studied and a bifurcation analysis of the steady states is fulfilled. Depending on the different parameters of the problem, we show that Hopf bifurcation may occur leading to oscillating behaviours of the system. The mathematical results are finally discussed.

Global Hopf bifurcation analysis on a BAM neural network with delays

NASA Astrophysics Data System (ADS)

Sun, Chengjun; Han, Maoan; Pang, Xiaoming

2007-01-01

A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large.

Using delay differential equations to induce alternans in a model of cardiac electrophysiology.

PubMed

Eastman, Justin; Sass, Julian; Gomes, Johnny M; Dos Santos, Rodrigo Weber; Cherry, Elizabeth M

2016-09-07

Cardiac electrical alternans is a period-2 dynamical behavior with alternating long and short action potential durations (APD) that often precedes dangerous arrhythmias associated with cardiac arrest. Despite the importance of alternans, many current ordinary differential equations models of cardiac electrophysiology do not produce alternans, thereby limiting the use of these models for studying the mechanisms that underlie this condition. Because delay differential equations (DDEs) commonly induce complex dynamics in other biological systems, we investigate whether incorporating DDEs can lead to alternans development in cardiac models by studying the Fox et al. canine ventricular action potential model. After suppressing the alternans in the original model, we show that alternans can be obtained by introducing DDEs in the model gating variables, and we quantitatively compare the DDE-induced alternans with the alternans present in the original model. We analyze the behavior of the voltage, currents, and gating variables of the model to study the effects of the delays and to determine how alternans develops in that setting, and we discuss the mathematical and physiological implications of our findings. In future work, we aim to apply our approach to induce alternans in models that do not naturally exhibit such dynamics. Copyright © 2016 Elsevier Ltd. All rights reserved.

Approximating a nonlinear advanced-delayed equation from acoustics

NASA Astrophysics Data System (ADS)

Teodoro, M. Filomena

2016-10-01

We approximate the solution of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.

Approximating a retarded-advanced differential equation that models human phonation

NASA Astrophysics Data System (ADS)

Teodoro, M. Filomena

2017-11-01

In [1, 2, 3] we have got the numerical solution of a linear mixed type functional differential equation (MTFDE) introduced initially in [4], considering the autonomous and non-autonomous case by collocation, least squares and finite element methods considering B-splines basis set. The present work introduces a numerical scheme using least squares method (LSM) and Gaussian basis functions to solve numerically a nonlinear mixed type equation with symmetric delay and advance which models human phonation. The preliminary results are promising. We obtain an accuracy comparable with the previous results.

Time-delayed feedback control of diffusion in random walkers.

PubMed

Ando, Hiroyasu; Takehara, Kohta; Kobayashi, Miki U

2017-07-01

Time delay in general leads to instability in some systems, while specific feedback with delay can control fluctuated motion in nonlinear deterministic systems to a stable state. In this paper, we consider a stochastic process, i.e., a random walk, and observe its diffusion phenomenon with time-delayed feedback. As a result, the diffusion coefficient decreases with increasing delay time. We analytically illustrate this suppression of diffusion by using stochastic delay differential equations and justify the feasibility of this suppression by applying time-delayed feedback to a molecular dynamics model.

Differential Equations and Computational Simulations

DTIC Science & Technology

1999-06-18

divergence operator of a vector field, which can be defined in terms of the Levi - Civita connection. Let $(x, t) be the orbit passing through x g M...differential equations 31 Junping Chen and Dadi Yang The limit cycle of two species predator-prey model with general functional response > 34 S. S...analysis of two -species nonlinear competition system with periodic coefficients 286 X. H. Tang and J. S. Yu Oscillation of first order delay

Dynamical analysis of cigarette smoking model with a saturated incidence rate

NASA Astrophysics Data System (ADS)

Zeb, Anwar; Bano, Ayesha; Alzahrani, Ebraheem; Zaman, Gul

2018-04-01

In this paper, we consider a delayed smoking model in which the potential smokers are assumed to satisfy the logistic equation. We discuss the dynamical behavior of our proposed model in the form of Delayed Differential Equations (DDEs) and show conditions for asymptotic stability of the model in steady state. We also discuss the Hopf bifurcation analysis of considered model. Finally, we use the nonstandard finite difference (NSFD) scheme to show the results graphically with help of MATLAB.

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

NASA Astrophysics Data System (ADS)

Niu, Ben; Zhang, Jiaming; Wei, Junjie

2018-05-01

In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.

Power-spectral-density relationship for retarded differential equations

NASA Technical Reports Server (NTRS)

Barker, L. K.

1974-01-01

The power spectral density (PSD) relationship between input and output of a set of linear differential-difference equations of the retarded type with real constant coefficients and delays is discussed. The form of the PSD relationship is identical with that applicable to unretarded equations. Since the PSD relationship is useful if and only if the system described by the equations is stable, the stability must be determined before applying the PSD relationship. Since it is sometimes difficult to determine the stability of retarded equations, such equations are often approximated by simpler forms. It is pointed out that some common approximations can lead to erroneous conclusions regarding the stability of a system and, therefore, to the possibility of obtaining PSD results which are not valid.

Wavefronts for a global reaction-diffusion population model with infinite distributed delay

NASA Astrophysics Data System (ADS)

Weng, Peixuan; Xu, Zhiting

2008-09-01

We consider a global reaction-diffusion population model with infinite distributed delay which includes models of Nicholson's blowflies and hematopoiesis derived by Gurney, Mackey and Glass, respectively. The existence of monotone wavefronts is derived by using the abstract settings of functional differential equations and Schauder fixed point theory.

Robotic reactions: delay-induced patterns in autonomous vehicle systems.

PubMed

Orosz, Gábor; Moehlis, Jeff; Bullo, Francesco

2010-02-01

Fundamental design principles are presented for vehicle systems governed by autonomous cruise control devices. By analyzing the corresponding delay differential equations, it is shown that for any car-following model short-wavelength oscillations can appear due to robotic reaction times, and that there are tradeoffs between the time delay and the control gains. The analytical findings are demonstrated on an optimal velocity model using numerical continuation and numerical simulation.

Robotic reactions: Delay-induced patterns in autonomous vehicle systems

NASA Astrophysics Data System (ADS)

Orosz, Gábor; Moehlis, Jeff; Bullo, Francesco

2010-02-01

Fundamental design principles are presented for vehicle systems governed by autonomous cruise control devices. By analyzing the corresponding delay differential equations, it is shown that for any car-following model short-wavelength oscillations can appear due to robotic reaction times, and that there are tradeoffs between the time delay and the control gains. The analytical findings are demonstrated on an optimal velocity model using numerical continuation and numerical simulation.

Stabilizing skateboard speed-wobble with reflex delay.

PubMed

Varszegi, Balazs; Takacs, Denes; Stepan, Gabor; Hogan, S John

2016-08-01

A simple mechanical model of the skateboard-skater system is analysed, in which the effect of human control is considered by means of a linear proportional-derivative (PD) controller with delay. The equations of motion of this non-holonomic system are neutral delay-differential equations. A linear stability analysis of the rectilinear motion is carried out analytically. It is shown how to vary the control gains with respect to the speed of the skateboard to stabilize the uniform motion. The critical reflex delay of the skater is determined as the function of the speed. Based on this analysis, we present an explanation for the linear instability of the skateboard-skater system at high speed. Moreover, the advantages of standing ahead of the centre of the board are demonstrated from the viewpoint of reflex delay and control gain sensitivity. © 2016 The Author(s).

A hybrid approach to parameter identification of linear delay differential equations involving multiple delays

NASA Astrophysics Data System (ADS)

Marzban, Hamid Reza

2018-05-01

In this paper, we are concerned with the parameter identification of linear time-invariant systems containing multiple delays. The approach is based upon a hybrid of block-pulse functions and Legendre's polynomials. The convergence of the proposed procedure is established and an upper error bound with respect to the L2-norm associated with the hybrid functions is derived. The problem under consideration is first transformed into a system of algebraic equations. The least squares technique is then employed for identification of the desired parameters. Several multi-delay systems of varying complexity are investigated to evaluate the performance and capability of the proposed approximation method. It is shown that the proposed approach is also applicable to a class of nonlinear multi-delay systems. It is demonstrated that the suggested procedure provides accurate results for the desired parameters.

Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval

NASA Astrophysics Data System (ADS)

Shi, Xiangyun; Kuang, Yang; Makroglou, Athena; Mokshagundam, Sriprakash; Li, Jiaxu

2017-11-01

Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life style. Both diabetic population and health expenses grow rapidly according to American Diabetes Association. Detecting the potential onset of T2DM is an essential focal point in the research of diabetes mellitus. The intravenous glucose tolerance test (IVGTT) is an effective protocol to determine the insulin sensitivity, glucose effectiveness, and pancreatic β-cell functionality, through the analysis and parameter estimation of a proper differential equation model. Delay differential equations have been used to study the complex physiological phenomena including the glucose and insulin regulations. In this paper, we propose a novel approach to model the time delay in IVGTT modeling. This novel approach uses two parameters to simulate not only both discrete time delay and distributed time delay in the past interval, but also the time delay distributed in a past sub-interval. Normally, larger time delay, either a discrete or a distributed delay, will destabilize the system. However, we find that time delay over a sub-interval might not. We present analytically some basic model properties, which are desirable biologically and mathematically. We show that this relatively simple model provides good fit to fluctuating patient data sets and reveals some intriguing dynamics. Moreover, our numerical simulation results indicate that our model may remove the defect in well known Minimal Model, which often overestimates the glucose effectiveness index.

Time delay and noise explaining the behaviour of the cell growth in fermentation process

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ayuobi, Tawfiqullah; Rosli, Norhayati; Bahar, Arifah

2015-02-03

This paper proposes to investigate the interplay between time delay and external noise in explaining the behaviour of the microbial growth in batch fermentation process. Time delay and noise are modelled jointly via stochastic delay differential equations (SDDEs). The typical behaviour of cell concentration in batch fermentation process under this model is investigated. Milstein scheme is applied for solving this model numerically. Simulation results illustrate the effects of time delay and external noise in explaining the lag and stationary phases, respectively for the cell growth of fermentation process.

Time delay and noise explaining the behaviour of the cell growth in fermentation process

NASA Astrophysics Data System (ADS)

Ayuobi, Tawfiqullah; Rosli, Norhayati; Bahar, Arifah; Salleh, Madihah Md

2015-02-01

This paper proposes to investigate the interplay between time delay and external noise in explaining the behaviour of the microbial growth in batch fermentation process. Time delay and noise are modelled jointly via stochastic delay differential equations (SDDEs). The typical behaviour of cell concentration in batch fermentation process under this model is investigated. Milstein scheme is applied for solving this model numerically. Simulation results illustrate the effects of time delay and external noise in explaining the lag and stationary phases, respectively for the cell growth of fermentation process.

Advanced-Retarded Differential Equations in Quantum Photonic Systems

NASA Astrophysics Data System (ADS)

Alvarez-Rodriguez, Unai; Perez-Leija, Armando; Egusquiza, Iñigo L.; Gräfe, Markus; Sanz, Mikel; Lamata, Lucas; Szameit, Alexander; Solano, Enrique

2017-02-01

We propose the realization of photonic circuits whose dynamics is governed by advanced-retarded differential equations. Beyond their mathematical interest, these photonic configurations enable the implementation of quantum feedback and feedforward without requiring any intermediate measurement. We show how this protocol can be applied to implement interesting delay effects in the quantum regime, as well as in the classical limit. Our results elucidate the potential of the protocol as a promising route towards integrated quantum control systems on a chip.

Advanced-Retarded Differential Equations in Quantum Photonic Systems

PubMed Central

Alvarez-Rodriguez, Unai; Perez-Leija, Armando; Egusquiza, Iñigo L.; Gräfe, Markus; Sanz, Mikel; Lamata, Lucas; Szameit, Alexander; Solano, Enrique

2017-01-01

We propose the realization of photonic circuits whose dynamics is governed by advanced-retarded differential equations. Beyond their mathematical interest, these photonic configurations enable the implementation of quantum feedback and feedforward without requiring any intermediate measurement. We show how this protocol can be applied to implement interesting delay effects in the quantum regime, as well as in the classical limit. Our results elucidate the potential of the protocol as a promising route towards integrated quantum control systems on a chip. PMID:28230090

Bifurcation Analysis and Application for Impulsive Systems with Delayed Impulses

NASA Astrophysics Data System (ADS)

Church, Kevin E. M.; Liu, Xinzhi

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov-Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point.

Trajectory controllability of semilinear systems with multiple variable delays in control

DOE Office of Scientific and Technical Information (OSTI.GOV)

Klamka, Jerzy, E-mail: Jerzy.Klamka@polsl.pl, E-mail: Michal.Niezabitowski@polsl.pl; Niezabitowski, Michał, E-mail: Jerzy.Klamka@polsl.pl, E-mail: Michal.Niezabitowski@polsl.pl

In this paper, finite-dimensional dynamical control system described by semilinear differential state equation with multiple variable delays in control are considered. The concept of controllability we extend on trajectory controllability for systems with multiple point delays in control. Moreover, remarks and comments on the relationships between different concepts of controllability are presented. Finally, simple numerical example, which illustrates theoretical considerations is also given. The possible extensions are also proposed.

A delay differential equation model of follicle waves in women.

PubMed

Panza, Nicole M; Wright, Andrew A; Selgrade, James F

2016-01-01

This article presents a mathematical model for hormonal regulation of the menstrual cycle which predicts the occurrence of follicle waves in normally cycling women. Several follicles of ovulatory size that develop sequentially during one menstrual cycle are referred to as follicle waves. The model consists of 13 nonlinear, delay differential equations with 51 parameters. Model simulations exhibit a unique stable periodic cycle and this menstrual cycle accurately approximates blood levels of ovarian and pituitary hormones found in the biological literature. Numerical experiments illustrate that the number of follicle waves corresponds to the number of rises in pituitary follicle stimulating hormone. Modifications of the model equations result in simulations which predict the possibility of two ovulations at different times during the same menstrual cycle and, hence, the occurrence of dizygotic twins via a phenomenon referred to as superfecundation. Sensitive parameters are identified and bifurcations in model behaviour with respect to parameter changes are discussed. Studying follicle waves may be helpful for improving female fertility and for understanding some aspects of female reproductive ageing.

Analysis of nonlocal neural fields for both general and gamma-distributed connectivities

NASA Astrophysics Data System (ADS)

Hutt, Axel; Atay, Fatihcan M.

2005-04-01

This work studies the stability of equilibria in spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integro-differential equation includes synaptic and space-dependent transmission delay for both general and gamma-distributed synaptic connectivities. The latter connectivity type reveals infinite, finite, and vanishing self-connectivities. The work derives conditions for stationary and nonstationary instabilities for both kernel types. In addition, a nonlinear analysis for general kernels yields the order parameter equation of the Turing instability. To compare the results to findings for partial differential equations (PDEs), two typical PDE-types are derived from the examined model equation, namely the general reaction-diffusion equation and the Swift-Hohenberg equation. Hence, the discussed integro-differential equation generalizes these PDEs. In the case of the gamma-distributed kernels, the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges. As a novel finding, we obtain Turing instabilities in fields with local inhibition-lateral excitation, while wave instabilities occur in fields with local excitation and lateral inhibition. Numerical simulations support the analytical results.

Scalable analysis of nonlinear systems using convex optimization

NASA Astrophysics Data System (ADS)

Papachristodoulou, Antonis

In this thesis, we investigate how convex optimization can be used to analyze different classes of nonlinear systems at various scales algorithmically. The methodology is based on the construction of appropriate Lyapunov-type certificates using sum of squares techniques. After a brief introduction on the mathematical tools that we will be using, we turn our attention to robust stability and performance analysis of systems described by Ordinary Differential Equations. A general framework for constrained systems analysis is developed, under which stability of systems with polynomial, non-polynomial vector fields and switching systems, as well estimating the region of attraction and the L2 gain can be treated in a unified manner. We apply our results to examples from biology and aerospace. We then consider systems described by Functional Differential Equations (FDEs), i.e., time-delay systems. Their main characteristic is that they are infinite dimensional, which complicates their analysis. We first show how the complete Lyapunov-Krasovskii functional can be constructed algorithmically for linear time-delay systems. Then, we concentrate on delay-independent and delay-dependent stability analysis of nonlinear FDEs using sum of squares techniques. An example from ecology is given. The scalable stability analysis of congestion control algorithms for the Internet is investigated next. The models we use result in an arbitrary interconnection of FDE subsystems, for which we require that stability holds for arbitrary delays, network topologies and link capacities. Through a constructive proof, we develop a Lyapunov functional for FAST---a recently developed network congestion control scheme---so that the Lyapunov stability properties scale with the system size. We also show how other network congestion control schemes can be analyzed in the same way. Finally, we concentrate on systems described by Partial Differential Equations. We show that axially constant perturbations of the Navier-Stokes equations for Hagen-Poiseuille flow are globally stable, even though the background noise is amplified as R3 where R is the Reynolds number, giving a 'robust yet fragile' interpretation. We also propose a sum of squares methodology for the analysis of systems described by parabolic PDEs. We conclude this work with an account for future research.

Anticontrol of chaos in continuous-time systems via time-delay feedback.

PubMed

Wang, Xiao Fan; Chen, Guanrong; Yu, Xinghuo

2000-12-01

In this paper, a systematic design approach based on time-delay feedback is developed for anticontrol of chaos in a continuous-time system. This anticontrol method can drive a finite-dimensional, continuous-time, autonomous system from nonchaotic to chaotic, and can also enhance the existing chaos of an originally chaotic system. Asymptotic analysis is used to establish an approximate relationship between a time-delay differential equation and a discrete map. Anticontrol of chaos is then accomplished based on this relationship and the differential-geometry control theory. Several examples are given to verify the effectiveness of the methodology and to illustrate the systematic design procedure. (c) 2000 American Institute of Physics.

Numerical analysis for trajectory controllability of a coupled multi-order fractional delay differential system via the shifted Jacobi method

NASA Astrophysics Data System (ADS)

Priya, B. Ganesh; Muthukumar, P.

2018-02-01

This paper deals with the trajectory controllability for a class of multi-order fractional linear systems subject to a constant delay in state vector. The solution for the coupled fractional delay differential equation is established by the Mittag-Leffler function. The necessary and sufficient condition for the trajectory controllability is formulated and proved by the generalized Gronwall's inequality. The approximate trajectory for the proposed system is obtained through the shifted Jacobi operational matrix method. The numerical simulation of the approximate solution shows the theoretical results. Finally, some remarks and comments on the existing results of constrained controllability for the fractional dynamical system are also presented.

Chenciner bubbles and torus break-up in a periodically forced delay differential equation

NASA Astrophysics Data System (ADS)

Keane, A.; Krauskopf, B.

2018-06-01

We study a generic model for the interaction of negative delayed feedback and periodic forcing that was first introduced by Ghil et al (2008 Nonlinear Process. Geophys. 15 417–33) in the context of the El Niño Southern Oscillation climate system. This model takes the form of a delay differential equation and has been shown in previous work to be capable of producing complicated dynamics, which is organised by resonances between the external forcing and dynamics induced by feedback. For certain parameter values, we observe in simulations the sudden disappearance of (two-frequency dynamics on) tori. This can be explained by the folding of invariant tori and their associated resonance tongues. It is known that two smooth tori cannot simply meet and merge; they must actually break up in complicated bifurcation scenarios that are organised within so-called resonance bubbles first studied by Chenciner. We identify and analyse such a Chenciner bubble in order to understand the dynamics at folds of tori. We conduct a bifurcation analysis of the Chenciner bubble by means of continuation software and dedicated simulations, whereby some bifurcations involve tori and are detected in appropriate two-dimensional projections associated with Poincaré sections. We find close agreement between the observed bifurcation structure in the Chenciner bubble and a previously suggested theoretical picture. As far as we are aware, this is the first time the bifurcation structure associated with a Chenciner bubble has been analysed in a delay differential equation and, in fact, for a flow rather than an explicit map. Following our analysis, we briefly discuss the possible role of folding tori and Chenciner bubbles in the context of tipping.

Existence and stability of periodic solutions of quasi-linear Korteweg — de Vries equation

NASA Astrophysics Data System (ADS)

Glyzin, S. D.; Kolesov, A. Yu; Preobrazhenskaia, M. M.

2017-01-01

We consider the scalar nonlinear differential-difference equation with two delays, which models electrical activity of a neuron. Under some additional suppositions for this equation well known method of quasi-normal forms can be applied. Its essence lies in the formal normalization of the Poincare - Dulac obtaining quasi-normal form and the subsequent application of the theorems of conformity. In this case, the result of the application of quasi-normal forms is a countable system of differential-difference equations, which can be turned into a boundary value problem of the Korteweg - de Vries equation. The investigation of this boundary value problem allows us to draw a conclusion about the behaviour of the original equation. Namely, for a suitable choice of parameters in the framework of this equation is implemented buffer phenomenon consisting in the presence of the bifurcation mechanism for the birth of an arbitrarily large number of stable cycles.

Global cluster synchronization in nonlinearly coupled community networks with heterogeneous coupling delays.

PubMed

Tseng, Jui-Pin

2017-02-01

This investigation establishes the global cluster synchronization of complex networks with a community structure based on an iterative approach. The units comprising the network are described by differential equations, and can be non-autonomous and involve time delays. In addition, units in the different communities can be governed by different equations. The coupling configuration of the network is rather general. The coupling terms can be non-diffusive, nonlinear, asymmetric, and with heterogeneous coupling delays. Based on this approach, both delay-dependent and delay-independent criteria for global cluster synchronization are derived. We implement the present approach for a nonlinearly coupled neural network with heterogeneous coupling delays. Two numerical examples are given to show that neural networks can behave in a variety of new collective ways under the synchronization criteria. These examples also demonstrate that neural networks remain synchronized in spite of coupling delays between neurons across different communities; however, they may lose synchrony if the coupling delays between the neurons within the same community are too large, such that the synchronization criteria are violated. Copyright © 2016 Elsevier Ltd. All rights reserved.

Computing the Lyapunov spectrum of a dynamical system from an observed time series

NASA Technical Reports Server (NTRS)

Brown, Reggie; Bryant, Paul; Abarbanel, Henry D. I.

1991-01-01

The paper examines the problem of accurately determining, from an observed time series, the Liapunov exponents for the dynamical system generating the data. It is shown that, even with very large data sets, it is clearly advantageous to utilize local neighborhood-to-neighborhood mappings with higher-order Taylor series rather than just local linear maps. This procedure is demonstrated on the Henon and Ikeda maps of the plane itself, the Lorenz system of three ordinary differential equations, and the Mackey-Glass delay differential equation.

Analysis of an age structured model for tick populations subject to seasonal effects

NASA Astrophysics Data System (ADS)

Liu, Kaihui; Lou, Yijun; Wu, Jianhong

2017-08-01

We investigate an age-structured hyperbolic equation model by allowing the birth and death functions to be density dependent and periodic in time with the consideration of seasonal effects. By studying the integral form solution of this general hyperbolic equation obtained through the method of integration along characteristics, we give a detailed proof of the uniqueness and existence of the solution in light of the contraction mapping theorem. With additional biologically natural assumptions, using the tick population growth as a motivating example, we derive an age-structured model with time-dependent periodic maturation delays, which is quite different from the existing population models with time-independent maturation delays. For this periodic differential system with seasonal delays, the basic reproduction number R0 is defined as the spectral radius of the next generation operator. Then, we show the tick population tends to die out when R0 < 1 while remains persistent if R0 > 1. When there is no intra-specific competition among immature individuals due to the sufficient availability of immature tick hosts, the global stability of the positive periodic state for the whole model system of four delay differential equations can be obtained with the observation that a scalar subsystem for the adult stage size can be decoupled. The challenge for the proof of such a global stability result can be overcome by introducing a new phase space, based on which, a periodic solution semiflow can be defined which is eventually strongly monotone and strictly subhomogeneous.

Spacecraft stability and control using new techniques for periodic and time-delayed systems

NASA Astrophysics Data System (ADS)

NAzari, Morad

This dissertation addresses various problems in spacecraft stability and control using specialized theoretical and numerical techniques for time-periodic and time-delayed systems. First, the effects of energy dissipation are considered in the dual-spin spacecraft, where the damper masses in the platform (?) and the rotor (?) cause energy loss in the system. Floquet theory is employed to obtain stability charts for different relative spin rates of the subsystem [special characters omitted] with respect to the subsystem [special characters omitted]. Further, the stability and bifurcation of delayed feedback spin stabilization of a rigid spacecraft is investigated. The spin is stabilized about the principal axis of the intermediate moment of inertia using a simple delayed feedback control law. In particular, linear stability is analyzed via the exponential-polynomial characteristic equations and then the method of multiple scales is used to obtain the normal form of the Hopf bifurcation. Next, the dynamics of a rigid spacecraft with nonlinear delayed multi-actuator feedback control are studied, where a nonlinear feedback controller using an inverse dynamics approach is sought for the controlled system to have the desired linear delayed closed-loop dynamics (CLD). Later, three linear state feedback control strategies based on Chebyshev spectral collocation and the Lyapunov Floquet transformation (LFT) are explored for regulation control of linear periodic time delayed systems. First , a delayed feedback control law with discrete delay is implemented and the stability of the closed-loop response is investigated in the parameter space of available control gains using infinite-dimensional Floquet theory. Second, the delay differential equation (DDE) is discretized into a large set of ordinary differential equations (ODEs) using the Chebyshev spectral continuous time approximation (CSCTA) and delayed feedback with distributed delay is applied. The third strategy involves use of both CSCTA and the reduced Lyapunov Floquet transformation (RLFT) in order to design a non-delayed feedback control law. The delayed Mathieu equation is used as an illustrative example in which the closed-loop response and control effort are compared for all three control strategies. Finally, three example applications of control of time-periodic astrodynamic systems, i.e. formation flying control for an elliptic Keplerian chief orbit, body-fixed hovering control over a tumbling asteroid, and stationkeeping in Earth-Moon L1 halo orbits, are shown using versions of the control strategies introduced above. These applications employ a mixture of feedforward and non-delayed periodic-gain state feedback for tracking control of natural and non-natural motions in these systems. A major conclusion is that control effort is minimized by employing periodic-gain (rather than constant-gain) feedback control in such systems.

A delay differential equation model for dengue transmission with regular visits to a mosquito breeding site

NASA Astrophysics Data System (ADS)

Yaacob, Y.; Yeak, S. H.; Lim, R. S.; Soewono, E.

2015-03-01

Dengue disease has been known as one of widely transmitted vector-borne diseases which potentially affects millions of people throughout the world especially in tropical and sub-tropical countries. One of the main factors contributing in the complication of the transmission process is the mobility of people in which people may get infection in the places far from their home. Here we construct a delay differential equation model for dengue transmission in a closed population where regular visits of people to a mosquito breeding site out of their residency such as traditional market take place daily. Basic reproductive ratio of the system is obtained and depends on the ratio between the outgoing rates of susceptible human and infective human. It is shown that the increase of mobility with different variation of mobility rates may contribute to different level of basic reproductive ratio as well as different level of outbreaks.

Hopf-Pitchfork Bifurcation in a Symmetrically Conservative Two-Mass System with Delay

NASA Astrophysics Data System (ADS)

Sun, Ye; Zhang, Chunrui; Cai, Yuting

2018-06-01

A symmetrically conservative two-mass system with time delay is considered here. We analyse the influence of interaction coefficient and time delay on the Hopf-pitchfork bifurcation. The bifurcation diagrams and phase portraits are then obtained by computing the normal forms for the system in which, particularly, the unfolding form for case III is seldom given in delayed differential equations. Furthermore, we also find some interesting dynamical behaviours of the original system, such as the coexistence of two stable non-trivial equilibria and a pair of stable periodic orbits, which are verified both theoretically and numerically.

Synchronization in networks with heterogeneous coupling delays

NASA Astrophysics Data System (ADS)

Otto, Andreas; Radons, Günter; Bachrathy, Dániel; Orosz, Gábor

2018-01-01

Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.

Control-based method to identify underlying delays of a nonlinear dynamical system.

PubMed

Yu, Dongchuan; Frasca, Mattia; Liu, Fang

2008-10-01

We suggest several stationary state control-based delay identification methods which do not require any structural information about the controlled systems and are applicable to systems described by delayed ordinary differential equations. This proposed technique includes three steps: (i) driving a system to a steady state; (ii) perturbing the control signal for shifting the steady state; and (iii) identifying all delays by detecting the time that the system is abruptly drawn out of stationarity. Some aspects especially important for applications are discussed as well, including interaction delay identification, stationary state convergence speed, performance comparison, and the influence of noise on delay identification. Several examples are presented to illustrate the reliability and robustness of all delay identification methods suggested.

Modeling nuclear processes by Simulink

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rashid, Nahrul Khair Alang Md, E-mail: nahrul@iium.edu.my

2015-04-29

Modelling and simulation are essential parts in the study of dynamic systems behaviours. In nuclear engineering, modelling and simulation are important to assess the expected results of an experiment before the actual experiment is conducted or in the design of nuclear facilities. In education, modelling can give insight into the dynamic of systems and processes. Most nuclear processes can be described by ordinary or partial differential equations. Efforts expended to solve the equations using analytical or numerical solutions consume time and distract attention from the objectives of modelling itself. This paper presents the use of Simulink, a MATLAB toolbox softwaremore » that is widely used in control engineering, as a modelling platform for the study of nuclear processes including nuclear reactor behaviours. Starting from the describing equations, Simulink models for heat transfer, radionuclide decay process, delayed neutrons effect, reactor point kinetic equations with delayed neutron groups, and the effect of temperature feedback are used as examples.« less

A Discrete Events Delay Differential System Model for Transmission of Vancomycin-Resistant Enterococcus (VRE) in Hospitals

DTIC Science & Technology

2010-09-19

estimated directly form the surveillance data Infection control measures were implemented in the form of health care worker hand - hygiene before and after...hospital infections , is used to motivate possibilities of modeling nosocomial infec- tion dynamics. This is done in the context of hospital monitoring and...model development. Key Words: Delay equations, discrete events, nosocomial infection dynamics, surveil- lance data, inverse problems, parameter

Passivity analysis of memristor-based impulsive inertial neural networks with time-varying delays.

PubMed

Wan, Peng; Jian, Jigui

2018-03-01

This paper focuses on delay-dependent passivity analysis for a class of memristive impulsive inertial neural networks with time-varying delays. By choosing proper variable transformation, the memristive inertial neural networks can be rewritten as first-order differential equations. The memristive model presented here is regarded as a switching system rather than employing the theory of differential inclusion and set-value map. Based on matrix inequality and Lyapunov-Krasovskii functional method, several delay-dependent passivity conditions are obtained to ascertain the passivity of the addressed networks. In addition, the results obtained here contain those on the passivity for the addressed networks without impulse effects as special cases and can also be generalized to other neural networks with more complex pulse interference. Finally, one numerical example is presented to show the validity of the obtained results. Copyright © 2018 ISA. Published by Elsevier Ltd. All rights reserved.

Linear models of activation cascades: analytical solutions and coarse-graining of delayed signal transduction

PubMed Central

Desikan, Radhika

2016-01-01

Cellular signal transduction usually involves activation cascades, the sequential activation of a series of proteins following the reception of an input signal. Here, we study the classic model of weakly activated cascades and obtain analytical solutions for a variety of inputs. We show that in the special but important case of optimal gain cascades (i.e. when the deactivation rates are identical) the downstream output of the cascade can be represented exactly as a lumped nonlinear module containing an incomplete gamma function with real parameters that depend on the rates and length of the cascade, as well as parameters of the input signal. The expressions obtained can be applied to the non-identical case when the deactivation rates are random to capture the variability in the cascade outputs. We also show that cascades can be rearranged so that blocks with similar rates can be lumped and represented through our nonlinear modules. Our results can be used both to represent cascades in computational models of differential equations and to fit data efficiently, by reducing the number of equations and parameters involved. In particular, the length of the cascade appears as a real-valued parameter and can thus be fitted in the same manner as Hill coefficients. Finally, we show how the obtained nonlinear modules can be used instead of delay differential equations to model delays in signal transduction. PMID:27581482

A heterogenous Cournot duopoly with delay dynamics: Hopf bifurcations and stability switching curves

NASA Astrophysics Data System (ADS)

Pecora, Nicolò; Sodini, Mauro

2018-05-01

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy.

Finite-Dimensional Representations for Controlled Diffusions with Delay

DOE Office of Scientific and Technical Information (OSTI.GOV)

Federico, Salvatore, E-mail: salvatore.federico@unimi.it; Tankov, Peter, E-mail: tankov@math.univ-paris-diderot.fr

2015-02-15

We study stochastic delay differential equations (SDDE) where the coefficients depend on the moving averages of the state process. As a first contribution, we provide sufficient conditions under which the solution of the SDDE and a linear path functional of it admit a finite-dimensional Markovian representation. As a second contribution, we show how approximate finite-dimensional Markovian representations may be constructed when these conditions are not satisfied, and provide an estimate of the error corresponding to these approximations. These results are applied to optimal control and optimal stopping problems for stochastic systems with delay.

Linear or linearizable first-order delay ordinary differential equations and their Lie point symmetries

NASA Astrophysics Data System (ADS)

Dorodnitsyn, Vladimir A.; Kozlov, Roman; Meleshko, Sergey V.; Winternitz, Pavel

2018-05-01

A recent article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs, which have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional subalgebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone, and we present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.

Fitted Fourier-pseudospectral methods for solving a delayed reaction-diffusion partial differential equation in biology

NASA Astrophysics Data System (ADS)

Adam, A. M. A.; Bashier, E. B. M.; Hashim, M. H. A.; Patidar, K. C.

2017-07-01

In this work, we design and analyze a fitted numerical method to solve a reaction-diffusion model with time delay, namely, a delayed version of a population model which is an extension of the logistic growth (LG) equation for a food-limited population proposed by Smith [F.E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology 44 (1963) 651-663]. Seeing that the analytical solution (in closed form) is hard to obtain, we seek for a robust numerical method. The method consists of a Fourier-pseudospectral semi-discretization in space and a fitted operator implicit-explicit scheme in temporal direction. The proposed method is analyzed for convergence and we found that it is unconditionally stable. Illustrative numerical results will be presented at the conference.

A numerical scheme for singularly perturbed reaction-diffusion problems with a negative shift via numerov method

NASA Astrophysics Data System (ADS)

Dinesh Kumar, S.; Nageshwar Rao, R.; Pramod Chakravarthy, P.

2017-11-01

In this paper, we consider a boundary value problem for a singularly perturbed delay differential equation of reaction-diffusion type. We construct an exponentially fitted numerical method using Numerov finite difference scheme, which resolves not only the boundary layers but also the interior layers arising from the delay term. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

Investigation of a Nonlinear Control System

NASA Technical Reports Server (NTRS)

Flugge-Lotz, I; Taylor, C F; Lindberg, H E

1958-01-01

A discontinuous variation of coefficients of the differential equation describing the linear control system before nonlinear elements are added is studied in detail. The nonlinear feedback is applied to a second-order system. Simulation techniques are used to study performance of the nonlinear control system and to compare it with the linear system for a wide variety of inputs. A detailed quantitative study of the influence of relay delays and of a transport delay is presented.

A delay differential model of ENSO variability: parametric instability and the distribution of extremes

NASA Astrophysics Data System (ADS)

Ghil, M.; Zaliapin, I.; Thompson, S.

2008-05-01

We consider a delay differential equation (DDE) model for El-Niño Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform stability analyses of the model in the three-dimensional space of its physically relevant parameters. Our results illustrate the role of these three parameters: strength of seasonal forcing b, atmosphere-ocean coupling κ, and propagation period τ of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are separated by a sharp neutral curve in the (b, τ) plane at constant κ. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling κ increases. In the unstable regime, spontaneous transitions occur in the mean "temperature" (i.e., thermocline depth), period, and extreme annual values, for purely periodic, seasonal forcing. The model reproduces the Devil's bleachers characterizing other ENSO models, such as nonlinear, coupled systems of partial differential equations; some of the features of this behavior have been documented in general circulation models, as well as in observations. We expect, therefore, similar behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.

Estimation of nonlinear pilot model parameters including time delay.

NASA Technical Reports Server (NTRS)

Schiess, J. R.; Roland, V. R.; Wells, W. R.

1972-01-01

Investigation of the feasibility of using a Kalman filter estimator for the identification of unknown parameters in nonlinear dynamic systems with a time delay. The problem considered is the application of estimation theory to determine the parameters of a family of pilot models containing delayed states. In particular, the pilot-plant dynamics are described by differential-difference equations of the retarded type. The pilot delay, included as one of the unknown parameters to be determined, is kept in pure form as opposed to the Pade approximations generally used for these systems. Problem areas associated with processing real pilot response data are included in the discussion.

Modelling and control issues of dynamically substructured systems: adaptive forward prediction taken as an example

PubMed Central

Tu, Jia-Ying; Hsiao, Wei-De; Chen, Chih-Ying

2014-01-01

Testing techniques of dynamically substructured systems dissects an entire engineering system into parts. Components can be tested via numerical simulation or physical experiments and run synchronously. Additional actuator systems, which interface numerical and physical parts, are required within the physical substructure. A high-quality controller, which is designed to cancel unwanted dynamics introduced by the actuators, is important in order to synchronize the numerical and physical outputs and ensure successful tests. An adaptive forward prediction (AFP) algorithm based on delay compensation concepts has been proposed to deal with substructuring control issues. Although the settling performance and numerical conditions of the AFP controller are improved using new direct-compensation and singular value decomposition methods, the experimental results show that a linear dynamics-based controller still outperforms the AFP controller. Based on experimental observations, the least-squares fitting technique, effectiveness of the AFP compensation and differences between delay and ordinary differential equations are discussed herein, in order to reflect the fundamental issues of actuator modelling in relevant literature and, more specifically, to show that the actuator and numerical substructure are heterogeneous dynamic components and should not be collectively modelled as a homogeneous delay differential equation. PMID:25104902

Electrocardiogram classification using delay differential equations

NASA Astrophysics Data System (ADS)

Lainscsek, Claudia; Sejnowski, Terrence J.

2013-06-01

Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.

Incorporating time-delays in S-System model for reverse engineering genetic networks.

PubMed

Chowdhury, Ahsan Raja; Chetty, Madhu; Vinh, Nguyen Xuan

2013-06-18

In any gene regulatory network (GRN), the complex interactions occurring amongst transcription factors and target genes can be either instantaneous or time-delayed. However, many existing modeling approaches currently applied for inferring GRNs are unable to represent both these interactions simultaneously. As a result, all these approaches cannot detect important interactions of the other type. S-System model, a differential equation based approach which has been increasingly applied for modeling GRNs, also suffers from this limitation. In fact, all S-System based existing modeling approaches have been designed to capture only instantaneous interactions, and are unable to infer time-delayed interactions. In this paper, we propose a novel Time-Delayed S-System (TDSS) model which uses a set of delay differential equations to represent the system dynamics. The ability to incorporate time-delay parameters in the proposed S-System model enables simultaneous modeling of both instantaneous and time-delayed interactions. Furthermore, the delay parameters are not limited to just positive integer values (corresponding to time stamps in the data), but can also take fractional values. Moreover, we also propose a new criterion for model evaluation exploiting the sparse and scale-free nature of GRNs to effectively narrow down the search space, which not only reduces the computation time significantly but also improves model accuracy. The evaluation criterion systematically adapts the max-min in-degrees and also systematically balances the effect of network accuracy and complexity during optimization. The four well-known performance measures applied to the experimental studies on synthetic networks with various time-delayed regulations clearly demonstrate that the proposed method can capture both instantaneous and delayed interactions correctly with high precision. The experiments carried out on two well-known real-life networks, namely IRMA and SOS DNA repair network in Escherichia coli show a significant improvement compared with other state-of-the-art approaches for GRN modeling.

Sensitivity of Dynamical Systems to Banach Space Parameters

DTIC Science & Technology

2005-02-13

We consider general nonlinear dynamical systems in a Banach space with dependence on parameters in a second Banach space. An abstract theoretical ... framework for sensitivity equations is developed. An application to measure dependent delay differential systems arising in a class of HIV models is presented.

Error compensation for hybrid-computer solution of linear differential equations

NASA Technical Reports Server (NTRS)

Kemp, N. H.

1970-01-01

Z-transform technique compensates for digital transport delay and digital-to-analog hold. Method determines best values for compensation constants in multi-step and Taylor series projections. Technique also provides hybrid-calculation error compared to continuous exact solution, plus system stability properties.

Stable Numerical Approach for Fractional Delay Differential Equations

NASA Astrophysics Data System (ADS)

Singh, Harendra; Pandey, Rajesh K.; Baleanu, D.

2017-12-01

In this paper, we present a new stable numerical approach based on the operational matrix of integration of Jacobi polynomials for solving fractional delay differential equations (FDDEs). The operational matrix approach converts the FDDE into a system of linear equations, and hence the numerical solution is obtained by solving the linear system. The error analysis of the proposed method is also established. Further, a comparative study of the approximate solutions is provided for the test examples of the FDDE by varying the values of the parameters in the Jacobi polynomials. As in special case, the Jacobi polynomials reduce to the well-known polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third and (4) Chebyshev polynomial of fourth kind respectively. Maximum absolute error and root mean square error are calculated for the illustrated examples and presented in form of tables for the comparison purpose. Numerical stability of the presented method with respect to all four kind of polynomials are discussed. Further, the obtained numerical results are compared with some known methods from the literature and it is observed that obtained results from the proposed method is better than these methods.

Synchronization of an Inertial Neural Network With Time-Varying Delays and Its Application to Secure Communication.

PubMed

Lakshmanan, Shanmugam; Prakash, Mani; Lim, Chee Peng; Rakkiyappan, Rajan; Balasubramaniam, Pagavathigounder; Nahavandi, Saeid

2018-01-01

In this paper, synchronization of an inertial neural network with time-varying delays is investigated. Based on the variable transformation method, we transform the second-order differential equations into the first-order differential equations. Then, using suitable Lyapunov-Krasovskii functionals and Jensen's inequality, the synchronization criteria are established in terms of linear matrix inequalities. Moreover, a feedback controller is designed to attain synchronization between the master and slave models, and to ensure that the error model is globally asymptotically stable. Numerical examples and simulations are presented to indicate the effectiveness of the proposed method. Besides that, an image encryption algorithm is proposed based on the piecewise linear chaotic map and the chaotic inertial neural network. The chaotic signals obtained from the inertial neural network are utilized for the encryption process. Statistical analyses are provided to evaluate the effectiveness of the proposed encryption algorithm. The results ascertain that the proposed encryption algorithm is efficient and reliable for secure communication applications.

Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations.

PubMed

Zhang, Ling

2017-01-01

The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Numerical bifurcation analysis of immunological models with time delays

NASA Astrophysics Data System (ADS)

Luzyanina, Tatyana; Roose, Dirk; Bocharov, Gennady

2005-12-01

In recent years, a large number of mathematical models that are described by delay differential equations (DDEs) have appeared in the life sciences. To analyze the models' dynamics, numerical methods are necessary, since analytical studies can only give limited results. In turn, the availability of efficient numerical methods and software packages encourages the use of time delays in mathematical modelling, which may lead to more realistic models. We outline recently developed numerical methods for bifurcation analysis of DDEs and illustrate the use of these methods in the analysis of a mathematical model of human hepatitis B virus infection.

Structural Properties and Estimation of Delay Systems. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Kwong, R. H. S.

1975-01-01

Two areas in the theory of delay systems were studied: structural properties and their applications to feedback control, and optimal linear and nonlinear estimation. The concepts of controllability, stabilizability, observability, and detectability were investigated. The property of pointwise degeneracy of linear time-invariant delay systems is considered. Necessary and sufficient conditions for three dimensional linear systems to be made pointwise degenerate by delay feedback were obtained, while sufficient conditions for this to be possible are given for higher dimensional linear systems. These results were applied to obtain solvability conditions for the minimum time output zeroing control problem by delay feedback. A representation theorem is given for conditional moment functionals of general nonlinear stochastic delay systems, and stochastic differential equations are derived for conditional moment functionals satisfying certain smoothness properties.

D3-Equivariant coupled advertising oscillators model

NASA Astrophysics Data System (ADS)

Zhang, Chunrui; Zheng, Huifeng

2011-04-01

A ring of three coupled advertising oscillators with delay is considered. Using the symmetric functional differential equation theories, the multiple Hopf bifurcations of the equilibrium at the origin are demonstrated. The existence of multiple branches of bifurcating periodic solution is obtained. Numerical simulation supports our analysis results.

Class-A mode-locked lasers: Fundamental solutions

NASA Astrophysics Data System (ADS)

Kovalev, Anton V.; Viktorov, Evgeny A.

2017-11-01

We consider a delay differential equation (DDE) model for mode-locked operation in class-A semiconductor lasers containing both gain and absorber sections. The material processes are adiabatically eliminated as these are considered fast in comparison to the delay time for a long cavity device. We determine the steady states and analyze their bifurcations using DDE-BIFTOOL [Engelborghs et al., ACM Trans. Math. Software 28, 1 (2002)]. Multiple forms of coexistence, transformation, and hysteretic behavior of stable steady states and fundamental periodic regimes are discussed in bifurcation diagrams.

A new approach to approximating the linear quadratic optimal control law for hereditary systems with control delays

NASA Technical Reports Server (NTRS)

Milman, M. H.

1985-01-01

A factorization approach is presented for deriving approximations to the optimal feedback gain for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the feedback kernels.

DYNAMIC BEHAVIOR OF A DELAY-DIFFERENTIAL EQUATION MODEL FOR THE HORMONAL REGULATION OF THE MENSTRUAL CYCLE

EPA Science Inventory

During the menstrual cycle, pituitary hormones stimulate the growth and development of ovarian follicles and the release of an ovum to be fertilized. The ovarian follicles secrete hormones during the cycle that regulate the production of the pituitary hormones creating positi...

Pseudo almost periodic solutions to some systems of nonlinear hyperbolic second-order partial differential equations

NASA Astrophysics Data System (ADS)

Al-Islam, Najja Shakir

In this Dissertation, the existence of pseudo almost periodic solutions to some systems of nonlinear hyperbolic second-order partial differential equations is established. For that, (Al-Islam [4]) is first studied and then obtained under some suitable assumptions. That is, the existence of pseudo almost periodic solutions to a hyperbolic second-order partial differential equation with delay. The second-order partial differential equation (1) represents a mathematical model for the dynamics of gas absorption, given by uxt+a x,tux=Cx,t,u x,t , u0,t=4 t, 1 where a : [0, L] x RR , C : [0, L] x R x RR , and ϕ : RR are (jointly) continuous functions ( t being the greatest integer function) and L > 0. The results in this Dissertation generalize those of Poorkarimi and Wiener [22]. Secondly, a generalization of the above-mentioned system consisting of the non-linear hyperbolic second-order partial differential equation uxt+a x,tux+bx,t ut+cx,tu=f x,t,u, x∈ 0,L,t∈ R, 2 equipped with the boundary conditions ux,0 =40x, u0,t=u 0t, uxx,0=y 0x, x∈0,L, t∈R, 3 where a, b, c : [0, L ] x RR and f : [0, L] x R x RR are (jointly) continuous functions is studied. Under some suitable assumptions, the existence and uniqueness of pseudo almost periodic solutions to particular cases, as well as the general case of the second-order hyperbolic partial differential equation (2) are studied. The results of all studies contained within this text extend those obtained by Aziz and Meyers [6] in the periodic setting.

Local Stability of AIDS Epidemic Model Through Treatment and Vertical Transmission with Time Delay

NASA Astrophysics Data System (ADS)

Novi W, Cascarilla; Lestari, Dwi

2016-02-01

This study aims to explain stability of the spread of AIDS through treatment and vertical transmission model. Human with HIV need a time to positively suffer AIDS. The existence of a time, human with HIV until positively suffer AIDS can be delayed for a time so that the model acquired is the model with time delay. The model form is a nonlinear differential equation with time delay, SIPTA (susceptible-infected-pre AIDS-treatment-AIDS). Based on SIPTA model analysis results the disease free equilibrium point and the endemic equilibrium point. The disease free equilibrium point with and without time delay are local asymptotically stable if the basic reproduction number is less than one. The endemic equilibrium point will be local asymptotically stable if the time delay is less than the critical value of delay, unstable if the time delay is more than the critical value of delay, and bifurcation occurs if the time delay is equal to the critical value of delay.

Delay Differential Equation Models of Normal and Diseased Electrocardiograms

NASA Astrophysics Data System (ADS)

Lainscsek, Claudia; Sejnowski, Terrence J.

Time series analysis with nonlinear delay differential equations (DDEs) is a powerful tool since it reveals spectral as well as nonlinear properties of the underlying dynamical system. Here global DDE models are used to analyze electrocardiography recordings (ECGs) in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. To capture distinguishing features of the different data types the number of terms and delays in the model as well as the order of nonlinearity of the DDE model have to be selected. The DDE structure selection is done in a supervised way by selecting the DDE that best separates different data types. We analyzed 24 h of data from 15 young healthy subjects in normal sinus rhythm (NSR) of 15 congestive heart failure (CHF) patients as well as of 15 subjects suffering from atrial fibrillation (AF) selected from the Physionet database. For the analysis presented here we used 5 min non-overlapping data windows on the raw data without any artifact removal. For classification performance we used the Cohen Kappa coefficient computed directly from the confusion matrix. The overall classification performance of the three groups was around 72-99 % on the 5 min windows for the different approaches. For 2 h data windows the classification for all three groups was above 95%.

Synchronization of Reaction-Diffusion Neural Networks With Dirichlet Boundary Conditions and Infinite Delays.

PubMed

Sheng, Yin; Zhang, Hao; Zeng, Zhigang

2017-10-01

This paper is concerned with synchronization for a class of reaction-diffusion neural networks with Dirichlet boundary conditions and infinite discrete time-varying delays. By utilizing theories of partial differential equations, Green's formula, inequality techniques, and the concept of comparison, algebraic criteria are presented to guarantee master-slave synchronization of the underlying reaction-diffusion neural networks via a designed controller. Additionally, sufficient conditions on exponential synchronization of reaction-diffusion neural networks with finite time-varying delays are established. The proposed criteria herein enhance and generalize some published ones. Three numerical examples are presented to substantiate the validity and merits of the obtained theoretical results.

Influence of distributed delays on the dynamics of a generalized immune system cancerous cells interactions model

NASA Astrophysics Data System (ADS)

Piotrowska, M. J.; Bodnar, M.

2018-01-01

We present a generalisation of the mathematical models describing the interactions between the immune system and tumour cells which takes into account distributed time delays. For the analytical study we do not assume any particular form of the stimulus function describing the immune system reaction to presence of tumour cells but we only postulate its general properties. We analyse basic mathematical properties of the considered model such as existence and uniqueness of the solutions. Next, we discuss the existence of the stationary solutions and analytically investigate their stability depending on the forms of considered probability densities that is: Erlang, triangular and uniform probability densities separated or not from zero. Particular instability results are obtained for a general type of probability densities. Our results are compared with those for the model with discrete delays know from the literature. In addition, for each considered type of probability density, the model is fitted to the experimental data for the mice B-cell lymphoma showing mean square errors at the same comparable level. For estimated sets of parameters we discuss possibility of stabilisation of the tumour dormant steady state. Instability of this steady state results in uncontrolled tumour growth. In order to perform numerical simulation, following the idea of linear chain trick, we derive numerical procedures that allow us to solve systems with considered probability densities using standard algorithm for ordinary differential equations or differential equations with discrete delays.

Efficient Processing of Data for Locating Lightning Strikes

NASA Technical Reports Server (NTRS)

Medelius, Pedro J.; Starr, Stan

2003-01-01

Two algorithms have been devised to increase the efficiency of processing of data in lightning detection and ranging (LDAR) systems so as to enable the accurate location of lightning strikes in real time. In LDAR, the location of a lightning strike is calculated by solving equations for the differences among the times of arrival (DTOAs) of the lightning signals at multiple antennas as functions of the locations of the antennas and the speed of light. The most difficult part of the problem is computing the DTOAs from digitized versions of the signals received by the various antennas. One way (a time-domain approach) to determine the DTOAs is to compute cross-correlations among variously differentially delayed replicas of the digitized signals and to select, as the DTOAs, those differential delays that yield the maximum correlations. Another way (a frequency-domain approach) to determine the DTOAs involves the computation of cross-correlations among Fourier transforms of variously differentially phased replicas of the digitized signals, along with utilization of the relationship among phase difference, time delay, and frequency.

Dynamic reduction with applications to mathematical biology and other areas.

PubMed

Sacker, Robert J; Von Bremen, Hubertus F

2007-10-01

In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the 'reduced' equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping Τ of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping Τ is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.

Modeling the Kinetics of Root Gravireaction

NASA Astrophysics Data System (ADS)

Kondrachuk, Alexander V.; Starkov, Vyacheslav N.

2011-02-01

The known "sun-flower equation" (SFE), which was originally proposed to model root circumnutating, was used to describe the simplest tip root graviresponse. Two forms of the SFE (integro-differential and differential-delayed) were solved, analyzed and compared with each other. The numerical solutions of these equations were found to be matching with arbitrary accuracy. The analysis of the solutions focused on time-lag effects on the kinetics of tip root bending. The results of the modeling are in good correlation with an experiment at the initial stages of root tips graviresponse. Further development of the model calls for its systematic comparison with some specially designed experiments, which would include measuring the kinetics of root tip bending before gravistimulation over the period of time longer than the time lag.

A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems

NASA Technical Reports Server (NTRS)

Banks, H. T.; Ito, K.

1991-01-01

A hybrid method for computing the feedback gains in linear quadratic regulator problem is proposed. The method, which combines use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite-dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantages of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed, and numerical evidence of the efficacy of these ideas is presented.

A numerical algorithm for optimal feedback gains in high dimensional LQR problems

NASA Technical Reports Server (NTRS)

Banks, H. T.; Ito, K.

1986-01-01

A hybrid method for computing the feedback gains in linear quadratic regulator problems is proposed. The method, which combines the use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated so as to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantage of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed and numerical evidence of the efficacy of our ideas presented.

Bounded Linear Stability Analysis - A Time Delay Margin Estimation Approach for Adaptive Control

NASA Technical Reports Server (NTRS)

Nguyen, Nhan T.; Ishihara, Abraham K.; Krishnakumar, Kalmanje Srinlvas; Bakhtiari-Nejad, Maryam

2009-01-01

This paper presents a method for estimating time delay margin for model-reference adaptive control of systems with almost linear structured uncertainty. The bounded linear stability analysis method seeks to represent the conventional model-reference adaptive law by a locally bounded linear approximation within a small time window using the comparison lemma. The locally bounded linear approximation of the combined adaptive system is cast in a form of an input-time-delay differential equation over a small time window. The time delay margin of this system represents a local stability measure and is computed analytically by a matrix measure method, which provides a simple analytical technique for estimating an upper bound of time delay margin. Based on simulation results for a scalar model-reference adaptive control system, both the bounded linear stability method and the matrix measure method are seen to provide a reasonably accurate and yet not too conservative time delay margin estimation.

Stability analysis of gyroscopic systems with delay via decomposition

NASA Astrophysics Data System (ADS)

Aleksandrov, A. Yu.; Zhabko, A. P.; Chen, Y.

2018-05-01

A mechanical system describing by the second order linear differential equations with a positive parameter at the velocity forces and with time delay in the positional forces is studied. Using the decomposition method and Lyapunov-Krasovskii functionals, conditions are obtained under which from the asymptotic stability of two auxiliary first order subsystems it follows that, for sufficiently large values of the parameter, the original system is also asymptotically stable. Moreover, it is shown that the proposed approach can be applied to the stability investigation of linear gyroscopic systems with switched positional forces.

The Emergence and Propagation of a Phase Boundary in an Elastic Bar.

DTIC Science & Technology

1983-06-01

differential- delay equations, where the delay 1s unknown. We first present a short- time analysis of this system in order to describe the emergence and Initial...ft, which arise on x=0 at progressively earlier times ; see Fig. 8. This geometric state of affairs is described by the inequalities : TT^"^^WF...t))) < 0 , (6.7) so that e(s(t)~,t) and v(s(t)",t) are decreasing with time . We have already observed that the first Inequality In (6.2

Delay-induced Turing-like waves for one-species reaction-diffusion model on a network

NASA Astrophysics Data System (ADS)

Petit, Julien; Carletti, Timoteo; Asllani, Malbor; Fanelli, Duccio

2015-09-01

A one-species time-delay reaction-diffusion system defined on a complex network is studied. Traveling waves are predicted to occur following a symmetry-breaking instability of a homogeneous stationary stable solution, subject to an external nonhomogeneous perturbation. These are generalized Turing-like waves that materialize in a single-species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time-delayed differential equations. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz network and with the inclusion of the delay.

Climate models with delay differential equations

NASA Astrophysics Data System (ADS)

Keane, Andrew; Krauskopf, Bernd; Postlethwaite, Claire M.

2017-11-01

A fundamental challenge in mathematical modelling is to find a model that embodies the essential underlying physics of a system, while at the same time being simple enough to allow for mathematical analysis. Delay differential equations (DDEs) can often assist in this goal because, in some cases, only the delayed effects of complex processes need to be described and not the processes themselves. This is true for some climate systems, whose dynamics are driven in part by delayed feedback loops associated with transport times of mass or energy from one location of the globe to another. The infinite-dimensional nature of DDEs allows them to be sufficiently complex to reproduce realistic dynamics accurately with a small number of variables and parameters. In this paper, we review how DDEs have been used to model climate systems at a conceptual level. Most studies of DDE climate models have focused on gaining insights into either the global energy balance or the fundamental workings of the El Niño Southern Oscillation (ENSO) system. For example, studies of DDEs have led to proposed mechanisms for the interannual oscillations in sea-surface temperature that is characteristic of ENSO, the irregular behaviour that makes ENSO difficult to forecast and the tendency of El Niño events to occur near Christmas. We also discuss the tools used to analyse such DDE models. In particular, the recent development of continuation software for DDEs makes it possible to explore large regions of parameter space in an efficient manner in order to provide a "global picture" of the possible dynamics. We also point out some directions for future research, including the incorporation of non-constant delays, which we believe could improve the descriptive power of DDE climate models.

Climate models with delay differential equations.

PubMed

Keane, Andrew; Krauskopf, Bernd; Postlethwaite, Claire M

2017-11-01

A fundamental challenge in mathematical modelling is to find a model that embodies the essential underlying physics of a system, while at the same time being simple enough to allow for mathematical analysis. Delay differential equations (DDEs) can often assist in this goal because, in some cases, only the delayed effects of complex processes need to be described and not the processes themselves. This is true for some climate systems, whose dynamics are driven in part by delayed feedback loops associated with transport times of mass or energy from one location of the globe to another. The infinite-dimensional nature of DDEs allows them to be sufficiently complex to reproduce realistic dynamics accurately with a small number of variables and parameters. In this paper, we review how DDEs have been used to model climate systems at a conceptual level. Most studies of DDE climate models have focused on gaining insights into either the global energy balance or the fundamental workings of the El Niño Southern Oscillation (ENSO) system. For example, studies of DDEs have led to proposed mechanisms for the interannual oscillations in sea-surface temperature that is characteristic of ENSO, the irregular behaviour that makes ENSO difficult to forecast and the tendency of El Niño events to occur near Christmas. We also discuss the tools used to analyse such DDE models. In particular, the recent development of continuation software for DDEs makes it possible to explore large regions of parameter space in an efficient manner in order to provide a "global picture" of the possible dynamics. We also point out some directions for future research, including the incorporation of non-constant delays, which we believe could improve the descriptive power of DDE climate models.

Delay differential analysis of time series.

PubMed

Lainscsek, Claudia; Sejnowski, Terrence J

2015-03-01

Nonlinear dynamical system analysis based on embedding theory has been used for modeling and prediction, but it also has applications to signal detection and classification of time series. An embedding creates a multidimensional geometrical object from a single time series. Traditionally either delay or derivative embeddings have been used. The delay embedding is composed of delayed versions of the signal, and the derivative embedding is composed of successive derivatives of the signal. The delay embedding has been extended to nonuniform embeddings to take multiple timescales into account. Both embeddings provide information on the underlying dynamical system without having direct access to all the system variables. Delay differential analysis is based on functional embeddings, a combination of the derivative embedding with nonuniform delay embeddings. Small delay differential equation (DDE) models that best represent relevant dynamic features of time series data are selected from a pool of candidate models for detection or classification. We show that the properties of DDEs support spectral analysis in the time domain where nonlinear correlation functions are used to detect frequencies, frequency and phase couplings, and bispectra. These can be efficiently computed with short time windows and are robust to noise. For frequency analysis, this framework is a multivariate extension of discrete Fourier transform (DFT), and for higher-order spectra, it is a linear and multivariate alternative to multidimensional fast Fourier transform of multidimensional correlations. This method can be applied to short or sparse time series and can be extended to cross-trial and cross-channel spectra if multiple short data segments of the same experiment are available. Together, this time-domain toolbox provides higher temporal resolution, increased frequency and phase coupling information, and it allows an easy and straightforward implementation of higher-order spectra across time compared with frequency-based methods such as the DFT and cross-spectral analysis.

Theoretical analysis of cross-talking signals between counter-streaming electron beams in a vacuum tube oscillator

NASA Astrophysics Data System (ADS)

Shin, Y. M.; Ryskin, N. M.; Won, J. H.; Han, S. T.; Park, G. S.

2006-03-01

The basic theory of cross-talking signals between counter-streaming electron beams in a vacuum tube oscillator consisting of two two-cavity klystron amplifiers reversely coupled through input/output slots is theoretically investigated. Application of Kirchhoff's laws to the coupled equivalent RLC circuit model of the device provides four nonlinear coupled equations, which are the first-order time-delayed differential equations. Analytical solutions obtained through linearization of the equations provide oscillation frequencies and thresholds of four fundamental eigenstates, symmetric/antisymmetric 0/π modes. Time-dependent output signals are numerically analyzed with variation of the beam current, and a self-modulation mechanism and transition to chaos scenario are examined. The oscillator shows a much stronger multistability compared to a delayed feedback klystron oscillator owing to the competitions among more diverse eigenmodes. A fully developed chaos region also appears at a relatively lower beam current, ˜3.5Ist, compared to typical vacuum tube oscillators (10-100Ist), where Ist is a start-oscillation current.

Chaos and Robustness in a Single Family of Genetic Oscillatory Networks

PubMed Central

Fu, Daniel; Tan, Patrick; Kuznetsov, Alexey; Molkov, Yaroslav I.

2014-01-01

Genetic oscillatory networks can be mathematically modeled with delay differential equations (DDEs). Interpreting genetic networks with DDEs gives a more intuitive understanding from a biological standpoint. However, it presents a problem mathematically, for DDEs are by construction infinitely-dimensional and thus cannot be analyzed using methods common for systems of ordinary differential equations (ODEs). In our study, we address this problem by developing a method for reducing infinitely-dimensional DDEs to two- and three-dimensional systems of ODEs. We find that the three-dimensional reductions provide qualitative improvements over the two-dimensional reductions. We find that the reducibility of a DDE corresponds to its robustness. For non-robust DDEs that exhibit high-dimensional dynamics, we calculate analytic dimension lines to predict the dependence of the DDEs’ correlation dimension on parameters. From these lines, we deduce that the correlation dimension of non-robust DDEs grows linearly with the delay. On the other hand, for robust DDEs, we find that the period of oscillation grows linearly with delay. We find that DDEs with exclusively negative feedback are robust, whereas DDEs with feedback that changes its sign are not robust. We find that non-saturable degradation damps oscillations and narrows the range of parameter values for which oscillations exist. Finally, we deduce that natural genetic oscillators with highly-regular periods likely have solely negative feedback. PMID:24667178

Controlling Mackey-Glass chaos.

PubMed

Kiss, Gábor; Röst, Gergely

2017-11-01

The Mackey-Glass equation is the representative example of delay induced chaotic behavior. Here, we propose various control mechanisms so that otherwise erratic solutions are forced to converge to the positive equilibrium or to a periodic orbit oscillating around that equilibrium. We take advantage of some recent results of the delay differential literature, when a sufficiently large domain of the phase space has been shown to be attractive and invariant, where the system is governed by monotone delayed feedback and chaos is not possible due to some Poincaré-Bendixson type results. We systematically investigate what control mechanisms are suitable to drive the system into such a situation and prove that constant perturbation, proportional feedback control, Pyragas control, and state dependent delay control can all be efficient to control Mackey-Glass chaos with properly chosen control parameters.

Controlling Mackey-Glass chaos

NASA Astrophysics Data System (ADS)

Kiss, Gábor; Röst, Gergely

2017-11-01

The Mackey-Glass equation is the representative example of delay induced chaotic behavior. Here, we propose various control mechanisms so that otherwise erratic solutions are forced to converge to the positive equilibrium or to a periodic orbit oscillating around that equilibrium. We take advantage of some recent results of the delay differential literature, when a sufficiently large domain of the phase space has been shown to be attractive and invariant, where the system is governed by monotone delayed feedback and chaos is not possible due to some Poincaré-Bendixson type results. We systematically investigate what control mechanisms are suitable to drive the system into such a situation and prove that constant perturbation, proportional feedback control, Pyragas control, and state dependent delay control can all be efficient to control Mackey-Glass chaos with properly chosen control parameters.

Time and frequency domain analysis of sampled data controllers via mixed operation equations

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1981-01-01

Specification of the mathematical equations required to define the dynamic response of a linear continuous plant, subject to sampled data control, is complicated by the fact that the digital components of the control system cannot be modeled via linear ordinary differential equations. This complication can be overcome by introducing two new mathematical operations; namely, the operation of zero order hold and digial delay. It is shown that by direct utilization of these operations, a set of linear mixed operation equations can be written and used to define the dynamic response characteristics of the controlled system. It also is shown how these linear mixed operation equations lead, in an automatable manner, directly to a set of finite difference equations which are in a format compatible with follow on time and frequency domain analysis methods.

NUCLEAR REACTOR AS THE OBJECT OF CONTROL. AUTOMATIC CONTROL OF AIRCRAFT ENGINES . B.S. Voronkev Collection of Articles

DOE Office of Scientific and Technical Information (OSTI.GOV)

None

BS> The dynamics of a power reactor is treated in some detail. Although the reactor is described by a nonlinear differential equation of the seventh order, a two-group approximstion with prompt neutrons and one averaged group of delayed neutrons may be used. When the reactor is in equilibrium, the reactor equation may be linearized in two ways. The effects of positive and negative coefficients of tins of the reactor are discussed. The nonlinear character of the control rods is trested. (D.L.C.)

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

NASA Astrophysics Data System (ADS)

Indekeu, Joseph O.; Smets, Ruben

2017-08-01

Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.

Global asymptotic stability and hopf bifurcation for a blood cell production model.

PubMed

Crauste, Fabien

2006-04-01

We analyze the asymptotic stability of a nonlinear system of two differential equations with delay, describing the dynamics of blood cell produc- tion. This process takes place in the bone marrow, where stem cells differen- tiate throughout division in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells in the introduction of cells in cycle, we are led to study a characteristic equation with delay-dependent coefficients. We determine a necessary and sufficient condition for the global stability of the first steady state of our model, which describes the popula- tion's dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial positive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature.

Hopf Bifurcation Analysis of a Gene Regulatory Network Mediated by Small Noncoding RNA with Time Delays and Diffusion

NASA Astrophysics Data System (ADS)

Li, Chengxian; Liu, Haihong; Zhang, Tonghua; Yan, Fang

2017-12-01

In this paper, a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied. Choosing the sum of delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation. It is shown that the sum of delays can induce Hopf bifurcation and the diffusion incorporated into the system can effect the amplitude of periodic solutions. Furthermore, the spatially homogeneous periodic solution always exists and the spatially inhomogeneous periodic solution will arise when the diffusion coefficients of protein and mRNA are suitably small. Particularly, the small RNA diffusion coefficient is more robust and its effect on model is much less than protein and mRNA. Finally, the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, numerical simulations are carried out to illustrate our theoretical analysis.

A stochastic delay model for pricing debt and equity: Numerical techniques and applications

NASA Astrophysics Data System (ADS)

Tambue, Antoine; Kemajou Brown, Elisabeth; Mohammed, Salah

2015-01-01

Delayed nonlinear models for pricing corporate liabilities and European options were recently developed. Using self-financed strategy and duplication we were able to derive a Random Partial Differential Equation (RPDE) whose solutions describe the evolution of debt and equity values of a corporate in the last delay period interval in the accompanied paper (Kemajou et al., 2012) [14]. In this paper, we provide robust numerical techniques to solve the delayed nonlinear model for the corporate value, along with the corresponding RPDEs modeling the debt and equity values of the corporate. Using financial data from some firms, we forecast and compare numerical solutions from both the nonlinear delayed model and classical Merton model with the real corporate data. From this comparison, it comes up that in corporate finance the past dependence of the firm value process may be an important feature and therefore should not be ignored.

Traveling waves in a continuum model of 1D schools

NASA Astrophysics Data System (ADS)

Oza, Anand; Kanso, Eva; Shelley, Michael

2017-11-01

We construct and analyze a continuum model of a 1D school of flapping swimmers. Our starting point is a delay differential equation that models the interaction between a swimmer and its upstream neighbors' wakes, which is motivated by recent experiments in the Applied Math Lab at NYU. We coarse-grain the evolution equations and derive PDEs for the swimmer density and variables describing the upstream wake. We study the equations both analytically and numerically, and find that a uniform density of swimmers destabilizes into a traveling wave. Our model makes a number of predictions about the properties of such traveling waves, and sheds light on the role of hydrodynamics in mediating the structure of swimming schools.

Stability analysis of a liquid fuel annular combustion chamber. M.S. Thesis

NASA Technical Reports Server (NTRS)

Mcdonald, G. H.

1979-01-01

The problems of combustion instability in an annular combustion chamber are investigated. A modified Galerkin method was used to produce a set of modal amplitude equations from the general nonlinear partial differential acoustic wave equation. From these modal amplitude equations, the two variable perturbation method was used to develop a set of approximate equations of a given order of magnitude. These equations were modeled to show the effects of velocity sensitive combustion instabilities by evaluating the effects of certain parameters in the given set of equations. By evaluating these effects, parameters which cause instabilities to occur in the combustion chamber can be ascertained. It is assumed that in the annular combustion chamber, the liquid propellants are injected uniformly across the injector face, the combustion processes are distributed throughout the combustion chamber, and that no time delay occurs in the combustion processes.

Stochastic nonlinear time series forecasting using time-delay reservoir computers: performance and universality.

PubMed

Grigoryeva, Lyudmila; Henriques, Julie; Larger, Laurent; Ortega, Juan-Pablo

2014-07-01

Reservoir computing is a recently introduced machine learning paradigm that has already shown excellent performances in the processing of empirical data. We study a particular kind of reservoir computers called time-delay reservoirs that are constructed out of the sampling of the solution of a time-delay differential equation and show their good performance in the forecasting of the conditional covariances associated to multivariate discrete-time nonlinear stochastic processes of VEC-GARCH type as well as in the prediction of factual daily market realized volatilities computed with intraday quotes, using as training input daily log-return series of moderate size. We tackle some problems associated to the lack of task-universality for individually operating reservoirs and propose a solution based on the use of parallel arrays of time-delay reservoirs. Copyright © 2014 Elsevier Ltd. All rights reserved.

Stochastic hybrid delay population dynamics: well-posed models and extinction.

PubMed

Yuan, Chenggui; Mao, Xuerong; Lygeros, John

2009-01-01

Nonlinear differential equations have been used for decades for studying fluctuations in the populations of species, interactions of species with the environment, and competition and symbiosis between species. Over the years, the original non-linear models have been embellished with delay terms, stochastic terms and more recently discrete dynamics. In this paper, we investigate stochastic hybrid delay population dynamics (SHDPD), a very general class of population dynamics that comprises all of these phenomena. For this class of systems, we provide sufficient conditions to ensure that SHDPD have global positive, ultimately bounded solutions, a minimum requirement for a realistic, well-posed model. We then study the question of extinction and establish conditions under which an ecosystem modelled by SHDPD is doomed.

Sampled-data chain-observer design for a class of delayed nonlinear systems

NASA Astrophysics Data System (ADS)

Kahelras, M.; Ahmed-Ali, T.; Giri, F.; Lamnabhi-Lagarrigue, F.

2018-05-01

The problem of observer design is addressed for a class of triangular nonlinear systems with not-necessarily small delay and sampled output measurements. One more difficulty is that the system state matrix is dependent on the un-delayed output signal which is not accessible to measurement, making existing observers inapplicable. A new chain observer, composed of m elementary observers in series, is designed to compensate for output sampling and arbitrary large delays. The larger the time-delay the larger the number m. Each elementary observer includes an output predictor that is conceived to compensate for the effects of output sampling and a fractional delay. The predictors are defined by first-order ordinary differential equations (ODEs) much simpler than those of existing predictors which involve both output and state predictors. Using a small gain type analysis, sufficient conditions for the observer to be exponentially convergent are established in terms of the minimal number m of elementary observers and the maximum sampling interval.

Stability of the human respiratory control system. I. Analysis of a two-dimensional delay state-space model.

PubMed

Batzel, J J; Tran, H T

2000-07-01

A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [12]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [17] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.

Stability of the human respiratory control system. II. Analysis of a three-dimensional delay state-space model.

PubMed

Batzel, J J; Tran, H T

2000-07-01

A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [1]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [4] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.

Theoretical analysis of cross-talking signals between counter-streaming electron beams in a vacuum tube oscillator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shin, Y.M.; Ryskin, N.M.; Won, J.H.

The basic theory of cross-talking signals between counter-streaming electron beams in a vacuum tube oscillator consisting of two two-cavity klystron amplifiers reversely coupled through input/output slots is theoretically investigated. Application of Kirchhoff's laws to the coupled equivalent RLC circuit model of the device provides four nonlinear coupled equations, which are the first-order time-delayed differential equations. Analytical solutions obtained through linearization of the equations provide oscillation frequencies and thresholds of four fundamental eigenstates, symmetric/antisymmetric 0/{pi} modes. Time-dependent output signals are numerically analyzed with variation of the beam current, and a self-modulation mechanism and transition to chaos scenario are examined. The oscillatormore » shows a much stronger multistability compared to a delayed feedback klystron oscillator owing to the competitions among more diverse eigenmodes. A fully developed chaos region also appears at a relatively lower beam current, {approx}3.5I{sub st}, compared to typical vacuum tube oscillators (10-100I{sub st}), where I{sub st} is a start-oscillation current.« less

Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks.

PubMed

Wang, Zhen; Campbell, Sue Ann

2017-11-01

We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with Z N symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.

Learning monopolies with delayed feedback on price expectations

NASA Astrophysics Data System (ADS)

Matsumoto, Akio; Szidarovszky, Ferenc

2015-11-01

We call the intercept of the price function with the vertical axis the maximum price and the slope of the price function the marginal price. In this paper it is assumed that a monopolistic firm has full information about the marginal price and its own cost function but is uncertain on the maximum price. However, by repeated interaction with the market, the obtained price observations give a basis for an adaptive learning process of the maximum price. It is also assumed that the price observations have fixed delays, so the learning process can be described by a delayed differential equation. In the cases of one or two delays, the asymptotic behavior of the resulting dynamic process is examined, stability conditions are derived. Three main results are demonstrated in the two delay learning processes. First, it is possible to stabilize the equilibrium which is unstable in the one delay model. Second, complex dynamics involving chaos, which is impossible in the one delay model, can emerge. Third, alternations of stability and instability (i.e., stability switches) occur repeatedly.

Stability and synchronization analysis of inertial memristive neural networks with time delays.

PubMed

Rakkiyappan, R; Premalatha, S; Chandrasekar, A; Cao, Jinde

2016-10-01

This paper is concerned with the problem of stability and pinning synchronization of a class of inertial memristive neural networks with time delay. In contrast to general inertial neural networks, inertial memristive neural networks is applied to exhibit the synchronization and stability behaviors due to the physical properties of memristors and the differential inclusion theory. By choosing an appropriate variable transmission, the original system can be transformed into first order differential equations. Then, several sufficient conditions for the stability of inertial memristive neural networks by using matrix measure and Halanay inequality are derived. These obtained criteria are capable of reducing computational burden in the theoretical part. In addition, the evaluation is done on pinning synchronization for an array of linearly coupled inertial memristive neural networks, to derive the condition using matrix measure strategy. Finally, the two numerical simulations are presented to show the effectiveness of acquired theoretical results.

Chaos based video encryption using maps and Ikeda time delay system

NASA Astrophysics Data System (ADS)

Valli, D.; Ganesan, K.

2017-12-01

Chaos based cryptosystems are an efficient method to deal with improved speed and highly secured multimedia encryption because of its elegant features, such as randomness, mixing, ergodicity, sensitivity to initial conditions and control parameters. In this paper, two chaos based cryptosystems are proposed: one is the higher-dimensional 12D chaotic map and the other is based on the Ikeda delay differential equation (DDE) suitable for designing a real-time secure symmetric video encryption scheme. These encryption schemes employ a substitution box (S-box) to diffuse the relationship between pixels of plain video and cipher video along with the diffusion of current input pixel with the previous cipher pixel, called cipher block chaining (CBC). The proposed method enhances the robustness against statistical, differential and chosen/known plain text attacks. Detailed analysis is carried out in this paper to demonstrate the security and uniqueness of the proposed scheme.

Application of Time-Delay Absorber to Suppress Vibration of a Dynamical System to Tuned Excitation.

PubMed

El-Ganaini, W A A; El-Gohary, H A

2014-08-01

In this work, we present a comprehensive investigation of the time delay absorber effects on the control of a dynamical system represented by a cantilever beam subjected to tuned excitation forces. Cantilever beam is one of the most widely used system in too many engineering applications, such as mechanical and civil engineering. The main aim of this work is to control the vibration of the beam at simultaneous internal and combined resonance condition, as it is the worst resonance case. Control is conducted via time delay absorber to suppress chaotic vibrations. Time delays often appear in many control systems in the state, in the control input, or in the measurements. Time delay commonly exists in various engineering, biological, and economical systems because of the finite speed of the information processing. It is a source of performance degradation and instability. Multiple time scale perturbation method is applied to obtain a first order approximation for the nonlinear differential equations describing the system behavior. The different resonance cases are reported and studied numerically. The stability of the steady-state solution at the selected worst resonance case is investigated applying Runge-Kutta fourth order method and frequency response equations via Matlab 7.0 and Maple11. Time delay absorber is effective, but within a specified range of time delay. It is the critical factor in selecting such absorber. Time delay absorber is better than the ordinary one as from the effectiveness point of view. The effects of the different absorber parameters on the system behavior and stability are studied numerically. A comparison with the available published work showed a close agreement with some previously published work.

Periodicity in cell dynamics in some mathematical models for the treatment of leukemia

NASA Astrophysics Data System (ADS)

Halanay, A.

2012-11-01

A model for the evolution of short-term hematopoietic stem cells and of leukocytes in leucemia under periodic treatment is introduced. It consists of a system of periodic delay differential equations and takes into consideration the asymmetric division. A guiding function is used, together with a theorem of Krasnoselskii, to prove the existence of a strictly positive periodic solution and its stability is investigated.

Modeling of long-range memory processes with inverse cubic distributions by the nonlinear stochastic differential equations

NASA Astrophysics Data System (ADS)

Kaulakys, B.; Alaburda, M.; Ruseckas, J.

2016-05-01

A well-known fact in the financial markets is the so-called ‘inverse cubic law’ of the cumulative distributions of the long-range memory fluctuations of market indicators such as a number of events of trades, trading volume and the logarithmic price change. We propose the nonlinear stochastic differential equation (SDE) giving both the power-law behavior of the power spectral density and the long-range dependent inverse cubic law of the cumulative distribution. This is achieved using the suggestion that when the market evolves from calm to violent behavior there is a decrease of the delay time of multiplicative feedback of the system in comparison to the driving noise correlation time. This results in a transition from the Itô to the Stratonovich sense of the SDE and yields a long-range memory process.

How time delay and network design shape response patterns in biochemical negative feedback systems.

PubMed

Börsch, Anastasiya; Schaber, Jörg

2016-08-24

Negative feedback in combination with time delay can bring about both sustained oscillations and adaptive behaviour in cellular networks. Here, we study which design features of systems with delayed negative feedback shape characteristic response patterns with special emphasis on the role of time delay. To this end, we analyse generic two-dimensional delay differential equations describing the dynamics of biochemical signal-response networks. We investigate the influence of several design features on the stability of the model equilibrium, i.e., presence of auto-inhibition and/or mass conservation and the kind and/or strength of the delayed negative feedback. We show that auto-inhibition and mass conservation have a stabilizing effect, whereas increasing abruptness and decreasing feedback threshold have a de-stabilizing effect on the model equilibrium. Moreover, applying our theoretical analysis to the mammalian p53 system we show that an auto-inhibitory feedback can decouple period and amplitude of an oscillatory response, whereas the delayed feedback can not. Our theoretical framework provides insight into how time delay and design features of biochemical networks act together to elicit specific characteristic response patterns. Such insight is useful for constructing synthetic networks and controlling their behaviour in response to external stimulation.

Homeostatic plasticity for single node delay-coupled reservoir computing.

PubMed

Toutounji, Hazem; Schumacher, Johannes; Pipa, Gordon

2015-06-01

Supplementing a differential equation with delays results in an infinite-dimensional dynamical system. This property provides the basis for a reservoir computing architecture, where the recurrent neural network is replaced by a single nonlinear node, delay-coupled to itself. Instead of the spatial topology of a network, subunits in the delay-coupled reservoir are multiplexed in time along one delay span of the system. The computational power of the reservoir is contingent on this temporal multiplexing. Here, we learn optimal temporal multiplexing by means of a biologically inspired homeostatic plasticity mechanism. Plasticity acts locally and changes the distances between the subunits along the delay, depending on how responsive these subunits are to the input. After analytically deriving the learning mechanism, we illustrate its role in improving the reservoir's computational power. To this end, we investigate, first, the increase of the reservoir's memory capacity. Second, we predict a NARMA-10 time series, showing that plasticity reduces the normalized root-mean-square error by more than 20%. Third, we discuss plasticity's influence on the reservoir's input-information capacity, the coupling strength between subunits, and the distribution of the readout coefficients.

Nonreciprocal wave scattering on nonlinear string-coupled oscillators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lepri, Stefano, E-mail: stefano.lepri@isc.cnr.it; Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino; Pikovsky, Arkady

2014-12-01

We study scattering of a periodic wave in a string on two lumped oscillators attached to it. The equations can be represented as a driven (by the incident wave) dissipative (due to radiation losses) system of delay differential equations of neutral type. Nonlinearity of oscillators makes the scattering non-reciprocal: The same wave is transmitted differently in two directions. Periodic regimes of scattering are analyzed approximately, using amplitude equation approach. We show that this setup can act as a nonreciprocal modulator via Hopf bifurcations of the steady solutions. Numerical simulations of the full system reveal nontrivial regimes of quasiperiodic and chaoticmore » scattering. Moreover, a regime of a “chaotic diode,” where transmission is periodic in one direction and chaotic in the opposite one, is reported.« less

Stability of a general delayed virus dynamics model with humoral immunity and cellular infection

NASA Astrophysics Data System (ADS)

Elaiw, A. M.; Raezah, A. A.; Alofi, A. S.

2017-06-01

In this paper, we investigate the dynamical behavior of a general nonlinear model for virus dynamics with virus-target and infected-target incidences. The model incorporates humoral immune response and distributed time delays. The model is a four dimensional system of delay differential equations where the production and removal rates of the virus and cells are given by general nonlinear functions. We derive the basic reproduction parameter R˜0 G and the humoral immune response activation number R˜1 G and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. We use suitable Lyapunov functionals and apply LaSalle's invariance principle to prove the global asymptotic stability of the all equilibria of the model. We confirm the theoretical results by numerical simulations.

Long-time behavior for suspension bridge equations with time delay

NASA Astrophysics Data System (ADS)

Park, Sun-Hye

2018-04-01

In this paper, we consider suspension bridge equations with time delay of the form u_{tt}(x,t) + Δ ^2 u (x,t) + k u^+ (x,t) + a_0 u_t (x,t) + a_1 u_t (x, t- τ ) + f(u(x,t)) = g(x). Many researchers have studied well-posedness, decay rates of energy, and existence of attractors for suspension bridge equations without delay effects. But, as far as we know, there is no work about suspension equations with time delay. In addition, there are not many studies on attractors for other delayed systems. Thus we first provide well-posedness for suspension equations with time delay. And then show the existence of global attractors and the finite dimensionality of the attractors by establishing energy functionals which are related to the norm of the phase space to our problem.

Stability and bifurcation in a model for the dynamics of stem-like cells in leukemia under treatment

NASA Astrophysics Data System (ADS)

Rǎdulescu, I. R.; Cândea, D.; Halanay, A.

2012-11-01

A mathematical model for the dynamics of leukemic cells during treatment is introduced. Delay differential equations are used to model cells' evolution and are based on the Mackey-Glass approach, incorporating Goldie-Coldman law. Since resistance is propagated by cells that have the capacity of self-renewal, a population of stem-like cells is studied. Equilibrium points are calculated and their stability properties are investigated.

Analysis of stability and bifurcations of fixed points and periodic solutions of a lumped model of neocortex with two delays

PubMed Central

2012-01-01

A lumped model of neural activity in neocortex is studied to identify regions of multi-stability of both steady states and periodic solutions. Presence of both steady states and periodic solutions is considered to correspond with epileptogenesis. The model, which consists of two delay differential equations with two fixed time lags is mainly studied for its dependency on varying connection strength between populations. Equilibria are identified, and using linear stability analysis, all transitions are determined under which both trivial and non-trivial fixed points lose stability. Periodic solutions arising at some of these bifurcations are numerically studied with a two-parameter bifurcation analysis. PMID:22655859

Parameter estimation and sensitivity analysis for a mathematical model with time delays of leukemia

NASA Astrophysics Data System (ADS)

Cândea, Doina; Halanay, Andrei; Rǎdulescu, Rodica; Tǎlmaci, Rodica

2017-01-01

We consider a system of nonlinear delay differential equations that describes the interaction between three competing cell populations: healthy, leukemic and anti-leukemia T cells involved in Chronic Myeloid Leukemia (CML) under treatment with Imatinib. The aim of this work is to establish which model parameters are the most important in the success or failure of leukemia remission under treatment using a sensitivity analysis of the model parameters. For the most significant parameters of the model which affect the evolution of CML disease during Imatinib treatment we try to estimate the realistic values using some experimental data. For these parameters, steady states are calculated and their stability is analyzed and biologically interpreted.

Function projective synchronization of memristor-based Cohen-Grossberg neural networks with time-varying delays.

PubMed

Abdurahman, Abdujelil; Jiang, Haijun; Rahman, Kaysar

2015-12-01

This paper deals with the problem of function projective synchronization for a class of memristor-based Cohen-Grossberg neural networks with time-varying delays. Based on the theory of differential equations with discontinuous right-hand side, some novel criteria are obtained to realize the function projective synchronization of addressed networks by combining open loop control and linear feedback control. As some special cases, several control strategies are given to ensure the realization of complete synchronization, anti-synchronization and the stabilization of the considered memristor-based Cohen-Grossberg neural network. Finally, a numerical example and its simulations are provided to demonstrate the effectiveness of the obtained results.

Movement of Landslide Triggered by Bedrock Exfiltration with Nonuniform Pore Pressure Distribution

NASA Astrophysics Data System (ADS)

Jan, C. D.; Jian, Z. K.

2014-12-01

Landslides are common phenomena of sediment movement in mountain areas and usually pose severe risks to people and infrastructure around those areas. The occurrence of landslides is influenced by groundwater dynamics and bedrock characteristics as well as by rainfall and soil-mass properties. The bedrock may drain or contribute to groundwater in the overlying soil mass, depending on the hydraulic conductivity, degree of fracturing, saturation, and hydraulic head. Our study here is based on the model proposed by Iverson (2005). The model describes the relation between landslide displacement and the shear-zone dilation/contraction of pore water pressure. To study landslide initiation and movement, a block soil mass sliding down an inclined beck-rock plane is governed by Newton's equation of motion, while both the bedrock exfiltration and excess pore pressure induced by dilatation or contraction of basal shear zone are described by diffusion equations. The Chebyshev collocation method was used to transform the governing equations to a system of first-order ordinary differential equations, without the need of iteration. Then a fourth-order Runge-Kutta scheme was used to solve these ordinary differential equations. The effects of nonuniform bedrock exfiltration pressure distributions, such as the delayed peak, central peak, and advanced peak distributions, on the time of landslide initiation and the speed of landslide movement were compared and discussed.

Stochastic resonance and noise delayed extinction in a model of two competing species

NASA Astrophysics Data System (ADS)

Valenti, D.; Fiasconaro, A.; Spagnolo, B.

2004-01-01

We study the role of the noise in the dynamics of two competing species. We consider generalized Lotka-Volterra equations in the presence of a multiplicative noise, which models the interaction between the species and the environment. The interaction parameter between the species is a random process which obeys a stochastic differential equation with a generalized bistable potential in the presence of a periodic driving term, which accounts for the environment temperature variation. We find noise-induced periodic oscillations of the species concentrations and stochastic resonance phenomenon. We find also a nonmonotonic behavior of the mean extinction time of one of the two competing species as a function of the additive noise intensity.

Parametric Identification of Nonlinear Dynamical Systems

NASA Technical Reports Server (NTRS)

Feeny, Brian

2002-01-01

In this project, we looked at the application of harmonic balancing as a tool for identifying parameters (HBID) in a nonlinear dynamical systems with chaotic responses. The main idea is to balance the harmonics of periodic orbits extracted from measurements of each coordinate during a chaotic response. The periodic orbits are taken to be approximate solutions to the differential equations that model the system, the form of the differential equations being known, but with unknown parameters to be identified. Below we summarize the main points addressed in this work. The details of the work are attached as drafts of papers, and a thesis, in the appendix. Our study involved the following three parts: (1) Application of the harmonic balance to a simulation case in which the differential equation model has known form for its nonlinear terms, in contrast to a differential equation model which has either power series or interpolating functions to represent the nonlinear terms. We chose a pendulum, which has sinusoidal nonlinearities; (2) Application of the harmonic balance to an experimental system with known nonlinear forms. We chose a double pendulum, for which chaotic response were easily generated. Thus we confronted a two-degree-of-freedom system, which brought forth challenging issues; (3) A study of alternative reconstruction methods. The reconstruction of the phase space is necessary for the extraction of periodic orbits from the chaotic responses, which is needed in this work. Also, characterization of a nonlinear system is done in the reconstructed phase space. Such characterizations are needed to compare models with experiments. Finally, some nonlinear prediction methods can be applied in the reconstructed phase space. We developed two reconstruction methods that may be considered if the common method (method of delays) is not applicable.

Neuron Learning to Network Organization.

DTIC Science & Technology

1983-12-20

02912 N 0-8 1t COTOLIGOF 1HV AflRS 12. REPORT OATE Pesne an ann Research Program December 20, 1983 Office of Naval Research , Code 442PT 13. NUMBER...visual cortc\\ from R. Cajal, Histologie du Systete Nerveux. mostly hard-wired and perform a great variety of control functions took hundreds of millions of...certain sense there is much that is known. A set of coupled non -linear differential equations. including time delays, can be written down that in

Modeling Protective Anti-Tumor Immunity via Preventative Cancer Vaccines Using a Hybrid Agent-based and Delay Differential Equation Approach

PubMed Central

Kim, Peter S.; Lee, Peter P.

2012-01-01

A next generation approach to cancer envisions developing preventative vaccinations to stimulate a person's immune cells, particularly cytotoxic T lymphocytes (CTLs), to eliminate incipient tumors before clinical detection. The purpose of our study is to quantitatively assess whether such an approach would be feasible, and if so, how many anti-cancer CTLs would have to be primed against tumor antigen to provide significant protection. To understand the relevant dynamics, we develop a two-compartment model of tumor-immune interactions at the tumor site and the draining lymph node. We model interactions at the tumor site using an agent-based model (ABM) and dynamics in the lymph node using a system of delay differential equations (DDEs). We combine the models into a hybrid ABM-DDE system and investigate dynamics over a wide range of parameters, including cell proliferation rates, tumor antigenicity, CTL recruitment times, and initial memory CTL populations. Our results indicate that an anti-cancer memory CTL pool of 3% or less can successfully eradicate a tumor population over a wide range of model parameters, implying that a vaccination approach is feasible. In addition, sensitivity analysis of our model reveals conditions that will result in rapid tumor destruction, oscillation, and polynomial rather than exponential decline in the tumor population due to tumor geometry. PMID:23133347

Predictive Control of the Blood Glucose Level in Type I Diabetic Patient Using Delay Differential Equation Wang Model.

PubMed

Esna-Ashari, Mojgan; Zekri, Maryam; Askari, Masood; Khalili, Noushin

2017-01-01

Because of increasing risk of diabetes, the measurement along with control of blood sugar has been of great importance in recent decades. In type I diabetes, because of the lack of insulin secretion, the cells cannot absorb glucose leading to low level of glucose. To control blood glucose (BG), the insulin must be injected to the body. This paper proposes a method for BG level regulation in type I diabetes. The control strategy is based on nonlinear model predictive control. The aim of the proposed controller optimized with genetics algorithms is to measure BG level each time and predict it for the next time interval. This merit causes a less amount of control effort, which is the rate of insulin delivered to the patient body. Consequently, this method can decrease the risk of hypoglycemia, a lethal phenomenon in regulating BG level in diabetes caused by a low BG level. Two delay differential equation models, namely Wang model and Enhanced Wang model, are applied as controller model and plant, respectively. The simulation results exhibit an acceptable performance of the proposed controller in meal disturbance rejection and robustness against parameter changes. As a result, if the nutrition of the person decreases instantly, the hypoglycemia will not happen. Furthermore, comparing this method with other works, it was shown that the new method outperforms previous studies.

Predictive Control of the Blood Glucose Level in Type I Diabetic Patient Using Delay Differential Equation Wang Model

PubMed Central

Esna-Ashari, Mojgan; Zekri, Maryam; Askari, Masood; Khalili, Noushin

2017-01-01

Because of increasing risk of diabetes, the measurement along with control of blood sugar has been of great importance in recent decades. In type I diabetes, because of the lack of insulin secretion, the cells cannot absorb glucose leading to low level of glucose. To control blood glucose (BG), the insulin must be injected to the body. This paper proposes a method for BG level regulation in type I diabetes. The control strategy is based on nonlinear model predictive control. The aim of the proposed controller optimized with genetics algorithms is to measure BG level each time and predict it for the next time interval. This merit causes a less amount of control effort, which is the rate of insulin delivered to the patient body. Consequently, this method can decrease the risk of hypoglycemia, a lethal phenomenon in regulating BG level in diabetes caused by a low BG level. Two delay differential equation models, namely Wang model and Enhanced Wang model, are applied as controller model and plant, respectively. The simulation results exhibit an acceptable performance of the proposed controller in meal disturbance rejection and robustness against parameter changes. As a result, if the nutrition of the person decreases instantly, the hypoglycemia will not happen. Furthermore, comparing this method with other works, it was shown that the new method outperforms previous studies. PMID:28487828

Assessing the efficiency of Wolbachia driven Aedes mosquito suppression by delay differential equations.

PubMed

Huang, Mugen; Luo, Jiaowan; Hu, Linchao; Zheng, Bo; Yu, Jianshe

2017-12-14

To suppress wild population of Aedes mosquitoes, the primary transmission vector of life-threatening diseases such as dengue, malaria, and Zika, an innovative strategy is to release male mosquitoes carrying the bacterium Wolbachia into natural areas to drive female sterility by cytoplasmic incompatibility. We develop a model of delay differential equations, incorporating the strong density restriction in the larval stage, to assess the delicate impact of life table parameters on suppression efficiency. Through mathematical analysis, we find the sufficient and necessary condition for global stability of the complete suppression state. This condition, combined with the experimental data for Aedes albopictus population in Guangzhou, helps us predict a large range of releasing intensities for suppression success. In particular, we find that if the number of released infected males is no less than four times the number of mosquitoes in wild areas, then the mosquito density in the peak season can be reduced by 95%. We introduce an index to quantify the dependence of suppression efficiency on parameters. The invariance of some quantitative properties of the index values under various perturbations of the same parameter justifies the applicability of this index, and the robustness of our modeling approach. The index yields a ranking of the sensitivity of all parameters, among which the adult mortality has the highest sensitivity and is considerably more sensitive than the natural larvae mortality. Copyright © 2017 Elsevier Ltd. All rights reserved.

Periodic solution of neutral Lotka-Volterra system with periodic delays

NASA Astrophysics Data System (ADS)

Liu, Zhijun; Chen, Lansun

2006-12-01

A nonautonomous n-species Lotka-Volterra system with neutral delays is investigated. A set of verifiable sufficient conditions is derived for the existence of at least one strictly positive periodic solution of this Lotka-Volterra system by applying an existence theorem and some analysis techniques, where the assumptions of the existence theorem are different from that of Gaines and Mawhin's continuation theorem [R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977] and that of abstract continuation theory for k-set contraction [W. Petryshyn, Z. Yu, Existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems, Nonlinear Anal. 6 (1982) 943-969]. Moreover, a problem proposed by Freedman and Wu [H.I. Freedman, J. Wu, Periodic solution of single species models with periodic delay, SIAM J. Math. Anal. 23 (1992) 689-701] is answered.

Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system.

PubMed

Hang, Chao; Huang, Guoxiang; Deng, L

2006-03-01

We investigate the influence of high-order dispersion and nonlinearity on the propagation of ultraslow optical solitons in a lifetime broadened four-state atomic system under a Raman excitation. Using a standard method of multiple-scales we derive a generalized nonlinear Schrödinger equation and show that for realistic physical parameters and at the pulse duration of 10(-6)s, the effects of third-order linear dispersion, nonlinear dispersion, and delay in nonlinear refractive index can be significant and may not be considered as perturbations. We provide exact soliton solutions for the generalized nonlinear Schrödinger equation and demonstrate that optical solitons obtained may still have ultraslow propagating velocity. Numerical simulations on the stability and interaction of these ultraslow optical solitons in the presence of linear and differential absorptions are also presented.

Fractional dynamics of globally slow transcription and its impact on deterministic genetic oscillation.

PubMed

Wei, Kun; Gao, Shilong; Zhong, Suchuan; Ma, Hong

2012-01-01

In dynamical systems theory, a system which can be described by differential equations is called a continuous dynamical system. In studies on genetic oscillation, most deterministic models at early stage are usually built on ordinary differential equations (ODE). Therefore, gene transcription which is a vital part in genetic oscillation is presupposed to be a continuous dynamical system by default. However, recent studies argued that discontinuous transcription might be more common than continuous transcription. In this paper, by appending the inserted silent interval lying between two neighboring transcriptional events to the end of the preceding event, we established that the running time for an intact transcriptional event increases and gene transcription thus shows slow dynamics. By globally replacing the original time increment for each state increment by a larger one, we introduced fractional differential equations (FDE) to describe such globally slow transcription. The impact of fractionization on genetic oscillation was then studied in two early stage models--the Goodwin oscillator and the Rössler oscillator. By constructing a "dual memory" oscillator--the fractional delay Goodwin oscillator, we suggested that four general requirements for generating genetic oscillation should be revised to be negative feedback, sufficient nonlinearity, sufficient memory and proper balancing of timescale. The numerical study of the fractional Rössler oscillator implied that the globally slow transcription tends to lower the chance of a coupled or more complex nonlinear genetic oscillatory system behaving chaotically.

Multi-scale modeling of the CD8 immune response

NASA Astrophysics Data System (ADS)

Barbarroux, Loic; Michel, Philippe; Adimy, Mostafa; Crauste, Fabien

2016-06-01

During the primary CD8 T-Cell immune response to an intracellular pathogen, CD8 T-Cells undergo exponential proliferation and continuous differentiation, acquiring cytotoxic capabilities to address the infection and memorize the corresponding antigen. After cleaning the organism, the only CD8 T-Cells left are antigen-specific memory cells whose role is to respond stronger and faster in case they are presented this very same antigen again. That is how vaccines work: a small quantity of a weakened pathogen is introduced in the organism to trigger the primary response, generating corresponding memory cells in the process, giving the organism a way to defend himself in case it encounters the same pathogen again. To investigate this process, we propose a non linear, multi-scale mathematical model of the CD8 T-Cells immune response due to vaccination using a maturity structured partial differential equation. At the intracellular scale, the level of expression of key proteins is modeled by a delay differential equation system, which gives the speeds of maturation for each cell. The population of cells is modeled by a maturity structured equation whose speeds are given by the intracellular model. We focus here on building the model, as well as its asymptotic study. Finally, we display numerical simulations showing the model can reproduce the biological dynamics of the cell population for both the primary response and the secondary responses.

Multi-scale modeling of the CD8 immune response

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barbarroux, Loic, E-mail: loic.barbarroux@doctorant.ec-lyon.fr; Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully; Michel, Philippe, E-mail: philippe.michel@ec-lyon.fr

During the primary CD8 T-Cell immune response to an intracellular pathogen, CD8 T-Cells undergo exponential proliferation and continuous differentiation, acquiring cytotoxic capabilities to address the infection and memorize the corresponding antigen. After cleaning the organism, the only CD8 T-Cells left are antigen-specific memory cells whose role is to respond stronger and faster in case they are presented this very same antigen again. That is how vaccines work: a small quantity of a weakened pathogen is introduced in the organism to trigger the primary response, generating corresponding memory cells in the process, giving the organism a way to defend himself inmore » case it encounters the same pathogen again. To investigate this process, we propose a non linear, multi-scale mathematical model of the CD8 T-Cells immune response due to vaccination using a maturity structured partial differential equation. At the intracellular scale, the level of expression of key proteins is modeled by a delay differential equation system, which gives the speeds of maturation for each cell. The population of cells is modeled by a maturity structured equation whose speeds are given by the intracellular model. We focus here on building the model, as well as its asymptotic study. Finally, we display numerical simulations showing the model can reproduce the biological dynamics of the cell population for both the primary response and the secondary responses.« less

Fisher-Wright model with deterministic seed bank and selection.

PubMed

Koopmann, Bendix; Müller, Johannes; Tellier, Aurélien; Živković, Daniel

2017-04-01

Seed banks are common characteristics to many plant species, which allow storage of genetic diversity in the soil as dormant seeds for various periods of time. We investigate an above-ground population following a Fisher-Wright model with selection coupled with a deterministic seed bank assuming the length of the seed bank is kept constant and the number of seeds is large. To assess the combined impact of seed banks and selection on genetic diversity, we derive a general diffusion model. The applied techniques outline a path of approximating a stochastic delay differential equation by an appropriately rescaled stochastic differential equation. We compute the equilibrium solution of the site-frequency spectrum and derive the times to fixation of an allele with and without selection. Finally, it is demonstrated that seed banks enhance the effect of selection onto the site-frequency spectrum while slowing down the time until the mutation-selection equilibrium is reached. Copyright © 2016 Elsevier Inc. All rights reserved.

Stability and oscillations in a CML model

NASA Astrophysics Data System (ADS)

Badralexi, Irina; Halanay, Andrei

2017-01-01

We capture the evolution in competition of healthy and leukemic cells in Chronic Myelogenous Leukemia (CML) taking into consideration the response of the immune system. Delay-differential equations in a Mackey-Glass approach are used. We start with the study of stability of the equilibrium points of the system. Conditions on parameters for the local stability are given. Oscillatory behaviors occur naturally in biological phenomena. Thus, we investigate the periodic behavior of solutions and we obtain conditions for periodic solutions to appear through a Hopf bifurcation.

Determinants of choice for pigeons and humans on concurrent-chains schedules of reinforcement.

PubMed

Belke, T W; Pierce, W D; Powell, R A

1989-09-01

Concurrent-chains schedules of reinforcement were arranged for humans and pigeons. Responses of humans were reinforced with tokens exchangeable for money, and key pecks of 4 birds were reinforced with food. Variable-interval 30-s and 40-s schedules operated in the terminal links of the chains. Condition 1 exposed subjects to variable-interval 90-s and variable-interval 30-s initial links, respectively. Conditions 2 and 3 arranged equal initial-link schedules of 40 s or 120 s. Experimental conditions tested the descriptive adequacy of five equations: reinforcement density, delay reduction, modified delay reduction, matching and maximization. Results based on choice proportions and switch rates during the initial links showed that pigeons behaved in accord with delay-reduction models, whereas humans maximized overall rate of reinforcement. As discussed by Logue and associates in self-control research, different types of reinforcement may affect sensitivity to delay differentially. Pigeons' responses were reinforced with food, a reinforcer that is consumable upon presentation. Humans' responses were reinforced with money, a reinforcer exchanged for consumable reinforcers after it was earned. Reinforcers that are immediately consumed may generate high sensitivity to delay and behavior described as delay reduction. Reinforces with longer times to consumption may generate low sensitivity to delay and behavior that maximizes overall payoff.

Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation

PubMed Central

Krauskopf, Bernd; Sieber, Jan

2014-01-01

Models of global climate phenomena of low to intermediate complexity are very useful for providing an understanding at a conceptual level. An important aspect of such models is the presence of a number of feedback loops that feature considerable delay times, usually due to the time it takes to transport energy (for example, in the form of hot/cold air or water) around the globe. In this paper, we demonstrate how one can perform a bifurcation analysis of the behaviour of a periodically forced system with delay in dependence on key parameters. As an example, we consider the El-Niño Southern Oscillation (ENSO), which is a sea-surface temperature (SST) oscillation on a multi-year scale in the basin of the Pacific Ocean. One can think of ENSO as being generated by an interplay between two feedback effects, one positive and one negative, which act only after some delay that is determined by the speed of transport of SST anomalies across the Pacific. We perform here a case study of a simple delayed-feedback oscillator model for ENSO, which is parametrically forced by annual variation. More specifically, we use numerical bifurcation analysis tools to explore directly regions of delay-induced resonances and other stability boundaries in this delay-differential equation model for ENSO. PMID:25197254

Theoretical and experimental study on multimode optical fiber grating

NASA Astrophysics Data System (ADS)

Yunming, Wang; Jingcao, Dai; Mingde, Zhang; Xiaohan, Sun

2005-06-01

The characteristics of multimode optical fiber Bragg grating (MMFBG) are studied theoretically and experimentally. For the first time the analysis of MMFBG based on a novel quasi-three-dimensional (Q-3D) finite-difference time-domain beam propagation method (Q-FDTD-BPM) is described through separating the angle component of vector field solution from the cylindrical coordinate so that several discrete two-dimensional (2D) equations are obtained, which simplify the 3D equations. And then these equations are developed using an alternating-direction implicit method and generalized Douglas scheme, which achieves higher accuracy than the regular FD scheme. All of the 2D solutions for the field intensities are also added with different power coefficients for different angle mode order numbers to obtain 3D field distributions in MMFBG. The presented method has been demonstrated as suitable simulation tool for analyzing MMFBG. In addition, based on the hydrogen-loaded and phase mask techniques, a series of Bragg grating have been written into the silicon multimode optical fiber loaded hydrogen for a month, and the spectrums for that have been measured, which obtain good results approximate to the results in the experiment. Group delay/differentiate group delay spectrums are obtained using Agilent 81910A Photonic All-Parameter Analyzer.

Time-delayed reaction-diffusion fronts

NASA Astrophysics Data System (ADS)

Isern, Neus; Fort, Joaquim

2009-11-01

A time-delayed second-order approximation for the front speed in reaction-dispersion systems was obtained by Fort and Méndez [Phys. Rev. Lett. 82, 867 (1999)]. Here we show that taking proper care of the effect of the time delay on the reactive process yields a different evolution equation and, therefore, an alternate equation for the front speed. We apply the new equation to the Neolithic transition. For this application the new equation yields speeds about 10% slower than the previous one.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

NASA Astrophysics Data System (ADS)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-01-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

NASA Astrophysics Data System (ADS)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-06-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0. Furthermore, we prove the global existence and uniqueness of C^{α ,β }-solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1-space. The exponential convergence rate is also derived.

Bifurcation and Control in a Singular Phytoplankton-Zooplankton-Fish Model with Nonlinear Fish Harvesting and Taxation

NASA Astrophysics Data System (ADS)

Meng, Xin-You; Wu, Yu-Qian

In this paper, a delayed differential algebraic phytoplankton-zooplankton-fish model with taxation and nonlinear fish harvesting is proposed. In the absence of time delay, the existence of singularity induced bifurcation is discussed by regarding economic interest as bifurcation parameter. A state feedback controller is designed to eliminate singularity induced bifurcation. Based on Liu’s criterion, Hopf bifurcation occurs at the interior equilibrium when taxation is taken as bifurcation parameter and is more than its corresponding critical value. In the presence of time delay, by analyzing the associated characteristic transcendental equation, the interior equilibrium loses local stability when time delay crosses its critical value. What’s more, the direction of Hopf bifurcation and stability of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem, and nonlinear state feedback controller is designed to eliminate Hopf bifurcation. Furthermore, Pontryagin’s maximum principle has been used to obtain optimal tax policy to maximize the benefit as well as the conservation of the ecosystem. Finally, some numerical simulations are given to demonstrate our theoretical analysis.

Finite-time synchronization for memristor-based neural networks with time-varying delays.

PubMed

Abdurahman, Abdujelil; Jiang, Haijun; Teng, Zhidong

2015-09-01

Memristive network exhibits state-dependent switching behaviors due to the physical properties of memristor, which is an ideal tool to mimic the functionalities of the human brain. In this paper, finite-time synchronization is considered for a class of memristor-based neural networks with time-varying delays. Based on the theory of differential equations with discontinuous right-hand side, several new sufficient conditions ensuring the finite-time synchronization of memristor-based chaotic neural networks are obtained by using analysis technique, finite time stability theorem and adding a suitable feedback controller. Besides, the upper bounds of the settling time of synchronization are estimated. Finally, a numerical example is given to show the effectiveness and feasibility of the obtained results. Copyright © 2015 Elsevier Ltd. All rights reserved.

Queues with Choice via Delay Differential Equations

NASA Astrophysics Data System (ADS)

Pender, Jamol; Rand, Richard H.; Wesson, Elizabeth

Delay or queue length information has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information. In the first fluid model, customers join each queue according to a Multinomial Logit Model, however, the queue length information the customer receives is delayed by a constant Δ. We show that the delay can cause oscillations or asynchronous behavior in the model based on the value of Δ. In the second model, customers receive information about the queue length through a moving average of the queue length. Although it has been shown empirically that giving patients moving average information causes oscillations and asynchronous behavior to occur in U.S. hospitals, we analytically and mathematically show for the first time that the moving average fluid model can exhibit oscillations and determine their dependence on the moving average window. Thus, our analysis provides new insight on how operators of service systems should report queue length information to customers and how delayed information can produce unwanted system dynamics.

Investigation of monolithic passively mode-locked quantum dot lasers with extremely low repetition frequency.

PubMed

Xu, Tianhong; Cao, Juncheng; Montrosset, Ivo

2015-01-01

The dynamical regimes and performance optimization of quantum dot monolithic passively mode-locked lasers with extremely low repetition rate are investigated using the numerical method. A modified multisection delayed differential equation model is proposed to accomplish simulations of both two-section and three-section passively mode-locked lasers with long cavity. According to the numerical simulations, it is shown that fundamental and harmonic mode-locking regimes can be multistable over a wide current range. These dynamic regimes are studied, and the reasons for their existence are explained. In addition, we demonstrate that fundamental pulses with higher peak power can be achieved when the laser is designed to work in a region with smaller differential gain.

A stochastically forced time delay solar dynamo model: Self-consistent recovery from a maunder-like grand minimum necessitates a mean-field alpha effect

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hazra, Soumitra; Nandy, Dibyendu; Passos, Dário, E-mail: s.hazra@iiserkol.ac.in, E-mail: dariopassos@ist.utl.pt, E-mail: dnandi@iiserkol.ac.in

Fluctuations in the Sun's magnetic activity, including episodes of grand minima such as the Maunder minimum have important consequences for space and planetary environments. However, the underlying dynamics of such extreme fluctuations remain ill-understood. Here, we use a novel mathematical model based on stochastically forced, non-linear delay differential equations to study solar cycle fluctuations in which time delays capture the physics of magnetic flux transport between spatially segregated dynamo source regions in the solar interior. Using this model, we explicitly demonstrate that the Babcock-Leighton poloidal field source based on dispersal of tilted bipolar sunspot flux, alone, cannot recover the sunspotmore » cycle from a grand minimum. We find that an additional poloidal field source effective on weak fields—e.g., the mean-field α effect driven by helical turbulence—is necessary for self-consistent recovery of the sunspot cycle from grand minima episodes.« less

Threshold Dynamics of a Temperature-Dependent Stage-Structured Mosquito Population Model with Nested Delays.

PubMed

Wang, Xiunan; Zou, Xingfu

2018-05-21

Mosquito-borne diseases remain a significant threat to public health and economics. Since mosquitoes are quite sensitive to temperature, global warming may not only worsen the disease transmission case in current endemic areas but also facilitate mosquito population together with pathogens to establish in new regions. Therefore, understanding mosquito population dynamics under the impact of temperature is considerably important for making disease control policies. In this paper, we develop a stage-structured mosquito population model in the environment of a temperature-controlled experiment. The model turns out to be a system of periodic delay differential equations with periodic delays. We show that the basic reproduction number is a threshold parameter which determines whether the mosquito population goes to extinction or remains persistent. We then estimate the parameter values for Aedes aegypti, the mosquito that transmits dengue virus. We verify the analytic result by numerical simulations with the temperature data of Colombo, Sri Lanka where a dengue outbreak occurred in 2017.

A simple and general method for solving detailed chemical evolution with delayed production of iron and other chemical elements

NASA Astrophysics Data System (ADS)

Vincenzo, F.; Matteucci, F.; Spitoni, E.

2017-04-01

We present a theoretical method for solving the chemical evolution of galaxies by assuming an instantaneous recycling approximation for chemical elements restored by massive stars and the delay time distribution formalism for delayed chemical enrichment by Type Ia Supernovae. The galaxy gas mass assembly history, together with the assumed stellar yields and initial mass function, represents the starting point of this method. We derive a simple and general equation, which closely relates the Laplace transforms of the galaxy gas accretion history and star formation history, which can be used to simplify the problem of retrieving these quantities in the galaxy evolution models assuming a linear Schmidt-Kennicutt law. We find that - once the galaxy star formation history has been reconstructed from our assumptions - the differential equation for the evolution of the chemical element X can be suitably solved with classical methods. We apply our model to reproduce the [O/Fe] and [Si/Fe] versus [Fe/H] chemical abundance patterns as observed at the solar neighbourhood by assuming a decaying exponential infall rate of gas and different delay time distributions for Type Ia Supernovae; we also explore the effect of assuming a non-linear Schmidt-Kennicutt law, with the index of the power law being k = 1.4. Although approximate, we conclude that our model with the single-degenerate scenario for Type Ia Supernovae provides the best agreement with the observed set of data. Our method can be used by other complementary galaxy stellar population synthesis models to predict also the chemical evolution of galaxies.

Predator-prey-subsidy population dynamics on stepping-stone domains with dispersal delays.

PubMed

Eide, Ragna M; Krause, Andrew L; Fadai, Nabil T; Van Gorder, Robert A

2018-08-14

We examine the role of the travel time of a predator along a spatial network on predator-prey population interactions, where the predator is able to partially or fully sustain itself on a resource subsidy. The impact of access to food resources on the stability and behaviour of the predator-prey-subsidy system is investigated, with a primary focus on how incorporating travel time changes the dynamics. The population interactions are modelled by a system of delay differential equations, where travel time is incorporated as discrete delay in the network diffusion term in order to model time taken to migrate between spatial regions. The model is motivated by the Arctic ecosystem, where the Arctic fox consumes both hunted lemming and scavenged seal carcass. The fox travels out on sea ice, in addition to quadrennially migrating over substantial distances. We model the spatial predator-prey-subsidy dynamics through a "stepping-stone" approach. We find that a temporal delay alone does not push species into extinction, but rather may stabilize or destabilize coexistence equilibria. We are able to show that delay can stabilize quasi-periodic or chaotic dynamics, and conclude that the incorporation of dispersal delay has a regularizing effect on dynamics, suggesting that dispersal delay can be proposed as a solution to the paradox of enrichment. Copyright © 2018 Elsevier Ltd. All rights reserved.

Comparison of two integration methods for dynamic causal modeling of electrophysiological data.

PubMed

Lemaréchal, Jean-Didier; George, Nathalie; David, Olivier

2018-06-01

Dynamic causal modeling (DCM) is a methodological approach to study effective connectivity among brain regions. Based on a set of observations and a biophysical model of brain interactions, DCM uses a Bayesian framework to estimate the posterior distribution of the free parameters of the model (e.g. modulation of connectivity) and infer architectural properties of the most plausible model (i.e. model selection). When modeling electrophysiological event-related responses, the estimation of the model relies on the integration of the system of delay differential equations (DDEs) that describe the dynamics of the system. In this technical note, we compared two numerical schemes for the integration of DDEs. The first, and standard, scheme approximates the DDEs (more precisely, the state of the system, with respect to conduction delays among brain regions) using ordinary differential equations (ODEs) and solves it with a fixed step size. The second scheme uses a dedicated DDEs solver with adaptive step sizes to control error, making it theoretically more accurate. To highlight the effects of the approximation used by the first integration scheme in regard to parameter estimation and Bayesian model selection, we performed simulations of local field potentials using first, a simple model comprising 2 regions and second, a more complex model comprising 6 regions. In these simulations, the second integration scheme served as the standard to which the first one was compared. Then, the performances of the two integration schemes were directly compared by fitting a public mismatch negativity EEG dataset with different models. The simulations revealed that the use of the standard DCM integration scheme was acceptable for Bayesian model selection but underestimated the connectivity parameters and did not allow an accurate estimation of conduction delays. Fitting to empirical data showed that the models systematically obtained an increased accuracy when using the second integration scheme. We conclude that inference on connectivity strength and delay based on DCM for EEG/MEG requires an accurate integration scheme. Copyright © 2018 The Authors. Published by Elsevier Inc. All rights reserved.

Stability of the mode-locking regime in tapered quantum-dot lasers

NASA Astrophysics Data System (ADS)

Bardella, P.; Drzewietzki, L.; Rossetti, M.; Weber, C.; Breuer, S.

2018-02-01

We study numerically and experimentally the role of the injection current and reverse bias voltage on the pulse stability of tapered, passively mode-locked, Quantum Dot (QD) lasers. By using a multi-section delayed differential equation and introducing in the model the QD inhomogenous broadening, we are able to predict the onset of leading and trailing edge instabilities in the emitted pulse trains and to identify specific trends of stability in dependence on the laser biasing conditions. The numerical results are confirmed experimentally trough amplitude and timing stability analysis of the pulses.

Dynamical modes of two almost identical chemical oscillators connected via both pulsatile and diffusive coupling.

PubMed

Safonov, Dmitry A; Vanag, Vladimir K

2018-05-03

The dynamical regimes of two almost identical Belousov-Zhabotinsky oscillators with both pulsatile (with time delay) and diffusive coupling have been studied theoretically with the aid of ordinary differential equations for four combinations of these types of coupling: inhibitory diffusive and inhibitory pulsatile (IDIP); excitatory diffusive and inhibitory pulsatile; inhibitory diffusive and excitatory pulsatile; and finally, excitatory diffusive and excitatory pulsatile (EDEP). The combination of two types of coupling creates a condition for new feedback, which promotes new dynamical modes for the IDIP and EDEP coupling.

Hyperbolic chaos in the klystron-type microwave vacuum tube oscillator

NASA Astrophysics Data System (ADS)

Emel'yanov, V. V.; Kuznetsov, S. P.; Ryskin, N. M.

2010-12-01

The ring-loop oscillator consisting of two coupled klystrons which is capable of generating hyperbolic chaotic signal in the microwave band is considered. The system of delayed-differential equations describing the dynamics of the oscillator is derived. This system is further reduced to the two-dimensional return map under the assumption of the instantaneous build-up of oscillations in the cavities. The results of detailed numerical simulation for both models are presented showing that there exists large enough range of control parameters where the sustained regime corresponds to the structurally stable hyperbolic chaos.

Spatial and temporal accuracy of asynchrony-tolerant finite difference schemes for partial differential equations at extreme scales

NASA Astrophysics Data System (ADS)

Kumari, Komal; Donzis, Diego

2017-11-01

Highly resolved computational simulations on massively parallel machines are critical in understanding the physics of a vast number of complex phenomena in nature governed by partial differential equations. Simulations at extreme levels of parallelism present many challenges with communication between processing elements (PEs) being a major bottleneck. In order to fully exploit the computational power of exascale machines one needs to devise numerical schemes that relax global synchronizations across PEs. This asynchronous computations, however, have a degrading effect on the accuracy of standard numerical schemes.We have developed asynchrony-tolerant (AT) schemes that maintain order of accuracy despite relaxed communications. We show, analytically and numerically, that these schemes retain their numerical properties with multi-step higher order temporal Runge-Kutta schemes. We also show that for a range of optimized parameters,the computation time and error for AT schemes is less than their synchronous counterpart. Stability of the AT schemes which depends upon history and random nature of delays, are also discussed. Support from NSF is gratefully acknowledged.

Dichotomies for generalized ordinary differential equations and applications

NASA Astrophysics Data System (ADS)

Bonotto, E. M.; Federson, M.; Santos, F. L.

2018-03-01

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.

Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

NASA Astrophysics Data System (ADS)

Chekroun, Mickaël D.; Glatt-Holtz, Nathan E.

2012-12-01

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space X which is acted on by any continuous semigroup { S( t)} t ≥ 0. Suppose that { S( t)} t ≥ 0 possesses a global attractor {{A}}. We show that, for any generalized Banach limit LIM T → ∞ and any probability distribution of initial conditions {{m}_0}, that there exists an invariant probability measure {{m}}, whose support is contained in {{A}}, such that intX \\varphi(x) d{m}(x) = \\underset{t rightarrow infty}LIM1/T int_0^T int_X \\varphi(S(t) x) d{m}_0(x) dt, for all observables φ living in a suitable function space of continuous mappings on X. This work is based on the framework of Foias et al. (Encyclopedia of mathematics and its applications, vol 83. Cambridge University Press, Cambridge, 2001); it generalizes and simplifies the proofs of more recent works (Wang in Disc Cont Dyn Syst 23(1-2):521-540, 2009; Lukaszewicz et al. in J Dyn Diff Eq 23(2):225-250, 2011). In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when { S( t)} t ≥ 0 does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and thus restricts the phase space X to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail. We first consider the Navier-Stokes equations with memory in the diffusion terms. This is the so called Jeffery's model which describes certain classes of viscoelastic fluids. We then consider a family of neutral delay differential equations, that is equations with delays in the time derivative terms. These systems may arise in the study of wave propagation problems coming from certain first order hyperbolic partial differential equations; for example for the study of line transmission problems. For the second example the phase space is {X= C([-tau,0],{R}^n)}, for some delay τ > 0, so that X is not reflexive in this case.

Finite-time synchronization of fractional-order memristor-based neural networks with time delays.

PubMed

Velmurugan, G; Rakkiyappan, R; Cao, Jinde

2016-01-01

In this paper, we consider the problem of finite-time synchronization of a class of fractional-order memristor-based neural networks (FMNNs) with time delays and investigated it potentially. By using Laplace transform, the generalized Gronwall's inequality, Mittag-Leffler functions and linear feedback control technique, some new sufficient conditions are derived to ensure the finite-time synchronization of addressing FMNNs with fractional order α:1<α<2 and 0<α<1. The results from the theory of fractional-order differential equations with discontinuous right-hand sides are used to investigate the problem under consideration. The derived results are extended to some previous related works on memristor-based neural networks. Finally, three numerical examples are presented to show the effectiveness of our proposed theoretical results. Copyright © 2015 Elsevier Ltd. All rights reserved.

Approximating the linear quadratic optimal control law for hereditary systems with delays in the control

NASA Technical Reports Server (NTRS)

Milman, Mark H.

1987-01-01

The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary systems. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.

Approximating the linear quadratic optimal control law for hereditary systems with delays in the control

NASA Technical Reports Server (NTRS)

Milman, Mark H.

1988-01-01

The fundamental control synthesis issue of establishing a priori convergence rates of approximation schemes for feedback controllers for a class of distributed parameter systems is addressed within the context of hereditary schemes. Specifically, a factorization approach is presented for deriving approximations to the optimal feedback gains for the linear regulator-quadratic cost problem associated with time-varying functional differential equations with control delays. The approach is based on a discretization of the state penalty which leads to a simple structure for the feedback control law. General properties of the Volterra factors of Hilbert-Schmidt operators are then used to obtain convergence results for the controls, trajectories and feedback kernels. Two algorithms are derived from the basic approximation scheme, including a fast algorithm, in the time-invariant case. A numerical example is also considered.

Global dynamics of a nonlocal delayed reaction-diffusion equation on a half plane

NASA Astrophysics Data System (ADS)

Hu, Wenjie; Duan, Yueliang

2018-04-01

We consider a delayed reaction-diffusion equation with spatial nonlocality on a half plane that describes population dynamics of a two-stage species living in a semi-infinite environment. A Neumann boundary condition is imposed accounting for an isolated domain. To describe the global dynamics, we first establish some a priori estimate for nontrivial solutions after investigating asymptotic properties of the nonlocal delayed effect and the diffusion operator, which enables us to show the permanence of the equation with respect to the compact open topology. We then employ standard dynamical system arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated by the diffusive Nicholson's blowfly equation and the diffusive Mackey-Glass equation.

Nonlinear Interaction of Detuned Instability Waves in Boundary-Layer Transition: Amplitude Equations

NASA Technical Reports Server (NTRS)

Lee, Sang Soo

1998-01-01

The non-equilibrium critical-layer analysis of a system of frequency-detuned resonant-triads is presented. In this part of the analysis, the system of partial differential critical-layer equations derived in Part I is solved analytically to yield the amplitude equations which are analyzed using a combination of asymptotic and numerical methods. Numerical solutions of the inviscid non-equilibrium oblique-mode amplitude equations show that the frequency-detuned self-interaction enhances the growth of the lower-frequency oblique modes more than the higher-frequency ones. All amplitudes become singular at the same finite downstream position. The frequency detuning delays the occurrence of the singularity. The spanwise-periodic mean-flow distortion and low-frequency nonlinear modes are generated by the critical-layer interaction between frequency-detuned oblique modes. The nonlinear mean flow and higher harmonics as well as the primary instabilities become as large as the base mean flow in the inviscid wall layer in the downstream region where the distance from the singularity is of the order of the wavelength scale.

[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (1)].

PubMed

Murase, Kenya

2014-01-01

Utilization of differential equations and methods for solving them in medical physics are presented. First, the basic concept and the kinds of differential equations were overviewed. Second, separable differential equations and well-known first-order and second-order differential equations were introduced, and the methods for solving them were described together with several examples. In the next issue, the symbolic and series expansion methods for solving differential equations will be mainly introduced.

The interaction evolution model of mass incidents with delay in a social network

NASA Astrophysics Data System (ADS)

Huo, Liang'an; Ma, Chenyang

2017-10-01

Recent years have witnessed rapid development of information technology. Today, modern media is widely used for the purpose of spreading information rapidly and widely. In particular, through micro-blog promotions, individuals tend to express their viewpoints and spread information on the internet, which could easily lead to public opinions. Moreover, government authorities also disseminate official information to guide public opinion and eliminate any incorrect conjecture. In this paper, a dynamical model with two delays is investigated to exhibit the interaction evolution between the public and official opinion fields in network mass incidents. Based on the theory of differential equations, the interaction mechanism between two public opinion fields in a micro-blog environment is analyzed. Two delays are proposed in the model to depict the response delays of public and official opinion fields. Some stable conditions are obtained, which shows that Hopf bifurcation can occur as delays cross critical values. Further, some numerical simulations are carried out to verify theoretical results. Our model indicates that there exists a golden time for government intervention, which should be emphasized given the impact of modern media and inaccurate rumors. If the government releases official information during the golden time, mass incidents on the internet can be controlled effectively.

Cycle-time equation for the Koller K300 cable yarder operating on steep slopes in the Northeast

Treesearch

Neil K. Huyler; Chris B. LeDoux

1997-01-01

Describes a delay-free-cycle time equation for the Koller K300 skyline yarder operating on steep slopes in the Northeast. Using the equation, the average delay-free-cycle time was 5.72 minutes. This means that about 420 cubic feet of material per hour can be produced. The important variables used in the equation were slope yarding distance, lateral yarding distance,...

[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (2)].

PubMed

Murase, Kenya

2015-01-01

In this issue, symbolic methods for solving differential equations were firstly introduced. Of the symbolic methods, Laplace transform method was also introduced together with some examples, in which this method was applied to solving the differential equations derived from a two-compartment kinetic model and an equivalent circuit model for membrane potential. Second, series expansion methods for solving differential equations were introduced together with some examples, in which these methods were used to solve Bessel's and Legendre's differential equations. In the next issue, simultaneous differential equations and various methods for solving these differential equations will be introduced together with some examples in medical physics.

Parallel Acquisition of Awareness and Differential Delay Eyeblink Conditioning

ERIC Educational Resources Information Center

Weidemann, Gabrielle; Antees, Cassandra

2012-01-01

There is considerable debate about whether differential delay eyeblink conditioning can be acquired without awareness of the stimulus contingencies. Previous investigations of the relationship between differential-delay eyeblink conditioning and awareness of the stimulus contingencies have assessed awareness after the conditioning session was…

On-line estimation and compensation of measurement delay in GPS/SINS integration

NASA Astrophysics Data System (ADS)

Yang, Tao; Wang, Wei

2008-10-01

The chief aim of this paper is to propose a simple on-line estimation and compensation method of GPS/SINS measurement delay. The causes of time delay for GPS/SINS integration are analyzed in this paper. New Kalman filter state equations augmented by measurement delay and modified measurement equations are derived. Based on an open-loop Kalman filter, several simulations are run, results of which show that by the proposed method, the estimation and compensation error of measurement delay is below 0.1s.

Teaching Modeling with Partial Differential Equations: Several Successful Approaches

ERIC Educational Resources Information Center

Myers, Joseph; Trubatch, David; Winkel, Brian

2008-01-01

We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…

The model of root graviresponse with retarded arguments

NASA Astrophysics Data System (ADS)

Kondrachuk, Alexander

The graviperception mechanism (GPM) of the roots of higher plants localized in the cap region of a root and supposedly related to statoliths sedimentation produces the signals in response to the change of the root axis orientation relative to the gravity vector G. Meanwhile, the regions (Distal Elongation Zone -DEZ and Central Elongation Zone-CEZ), where the signals initiate the changes of the growth rates of the upper and lower flanks of the root, are located at the significant distances from the cap (thousands microns for some plants). It causes the time delays between the relocation of statoliths in statocytes and the change of the growth rates in elongation zones. It is suggested that the signal targeting the CEZ modulates the initially uniform lateral distribution of some specific substances (S) in the cap region. Then already nonhomogeneous lateral distribution of S is transferred to the CEZ to initiate the change of the growth rates of the opposite flanks. It results in the bending of the root in the line of G and thus in the change of the GPM signal in the cap region. In the present model the kinetics of a root apex bending (angle A) in response to the time (t)-dependent change of the G orientation is described by the integro-differential equation in A(t). The main peculiarity of this model is the presence of retarded (time-delayed) arguments t-TCEZ and t-TDEZ . In this case the solutions of this equation depend on the preceding kinetics of A(t) during the time delays TCEZ and TDEZ . It is suggested that the signals activating the CEZ and DEZ are of different nature. The work is focused on two problems concerning the modeling of the effects of time-delay(s) on the root bending. The first problem supposes the existence of one zone (CEZ) and one time-delay TCEZ . This equation was studied and solved using analytical and numerical methods. We analyzed the model as to whether it can be used to describe the kinetics of root graviresponse in the case of different orientations of the root apex relative to the G vector during the time interval equal to TCEZ (TCEZ > TDEZ ) that precedes the beginning of gravistimultion. Also we explored the conditions of the overshooting (the vertical) and non-overshooting regimes of gravistimulated root bending. Good correlation between the results of the modeling and known experimental data (Barlow et al, 1993, Stochkus, 1994, Mullen, 1998) was found. This allowed us to estimate and analyze the parameters of the model. The second problem supposed the existence of two zones of growth (CEZ and DEZ) and two corresponding time-delays. The effects of the second time-delay connected with the presence of the DEZ on the behavior of the model equation of the root graviresponse kinetics were analyzed and discussed.

A dynamic IS-LM business cycle model with two time delays in capital accumulation equation

NASA Astrophysics Data System (ADS)

Zhou, Lujun; Li, Yaqiong

2009-06-01

In this paper, we analyze a augmented IS-LM business cycle model with the capital accumulation equation that two time delays are considered in investment processes according to Kalecki's idea. Applying stability switch criteria and Hopf bifurcation theory, we prove that time delays cause the equilibrium to lose or gain stability and Hopf bifurcation occurs.

Delay feedback induces a spontaneous motion of two-dimensional cavity solitons in driven semiconductor microcavities

NASA Astrophysics Data System (ADS)

Tlidi, M.; Averlant, E.; Vladimirov, A.; Panajotov, K.

2012-09-01

We consider a broad area vertical-cavity surface-emitting laser (VCSEL) operating below the lasing threshold and subject to optical injection and time-delayed feedback. We derive a generalized delayed Swift-Hohenberg equation for the VCSEL system, which is valid close to the nascent optical bistability. We first characterize the stationary-cavity solitons by constructing their snaking bifurcation diagram and by showing clustering behavior within the pinning region of parameters. Then, we show that the delayed feedback induces a spontaneous motion of two-dimensional (2D) cavity solitons in an arbitrary direction in the transverse plane. We characterize moving cavity solitons by estimating their threshold and calculating their velocity. Numerical 2D solutions of the governing semiconductor laser equations are in close agreement with those obtained from the delayed generalized Swift-Hohenberg equation.

On the integration of a class of nonlinear systems of ordinary differential equations

NASA Astrophysics Data System (ADS)

Talyshev, Aleksandr A.

2017-11-01

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

Delay Differential Equation-Based Modeling of Passively Mode-Locked Quantum Dot Lasers Using Measured Gain and Loss Spectra (Postprint)

DTIC Science & Technology

2013-01-01

semiconductor laser8, 9: ATAe d dA TQiTGi qg 12 11 2 1 (1) 2 0 1 AeeGgd dG GQ (2) Proc. of SPIE Vol. 8255 82551K-2 Downloaded From: http...1 Downloaded From: http://spiedigitallibrary.org/ on 01/14/2013 Terms of Use: http://spiedl.org/terms 1 Approved for public release; distribution...parameters ( 0g and 0q , respectively) have been derived to yield: Gg0 (10) Proc. of SPIE Vol. 8255 82551K-3 Downloaded From: http

Collaborative Research: Robust Climate Projections and Stochastic Stability of Dynamical Systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ilya Zaliapin

This project focused on conceptual exploration of El Nino/Southern Oscillation (ENSO) variability and sensitivity using a Delay Differential Equation developed in the project. We have (i) established the existence and continuous dependence of solutions of the model (ii) explored multiple models solutions, and the distribution of solutions extrema, and (iii) established and explored the phase locking phenomenon and the existence of multiple solutions for the same values of model parameters. In addition, we have applied to our model the concept of pullback attractor, which greatly facilitated predictive understanding of the nonlinear model's behavior.

External stimulation strength controls actin response dynamics in Dictyostelium cells

NASA Astrophysics Data System (ADS)

Hsu, Hsin-Fang; Westendorf, Christian; Tarantola, Marco; Zykov, Vladimir; Bodenschatz, Eberhard; Beta, Carsten

2015-03-01

Self-sustained oscillation and the resonance frequency of the cytoskeletal actin polymerization/depolymerization have recently been observed in Dictyostelium, a model system for studying chemotaxis. Here we report that the resonance frequency is not constant but rather varies with the strength of external stimuli. To understand the underlying mechanism, we analyzed the polymerization and depolymerization time at different levels of external stimulation. We found that polymerization time is independent of external stimuli but the depolymerization time is prolonged as the stimulation increases. These observations can be successfully reproduced in the frame work of our time delayed differential equation model.

Mathematical Methods for Physics and Engineering Third Edition Paperback Set

NASA Astrophysics Data System (ADS)

Riley, Ken F.; Hobson, Mike P.; Bence, Stephen J.

2006-06-01

Prefaces; 1. Preliminary algebra; 2. Preliminary calculus; 3. Complex numbers and hyperbolic functions; 4. Series and limits; 5. Partial differentiation; 6. Multiple integrals; 7. Vector algebra; 8. Matrices and vector spaces; 9. Normal modes; 10. Vector calculus; 11. Line, surface and volume integrals; 12. Fourier series; 13. Integral transforms; 14. First-order ordinary differential equations; 15. Higher-order ordinary differential equations; 16. Series solutions of ordinary differential equations; 17. Eigenfunction methods for differential equations; 18. Special functions; 19. Quantum operators; 20. Partial differential equations: general and particular; 21. Partial differential equations: separation of variables; 22. Calculus of variations; 23. Integral equations; 24. Complex variables; 25. Application of complex variables; 26. Tensors; 27. Numerical methods; 28. Group theory; 29. Representation theory; 30. Probability; 31. Statistics; Index.

Oscillation of a class of fractional differential equations with damping term.

PubMed

Qin, Huizeng; Zheng, Bin

2013-01-01

We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations

NASA Astrophysics Data System (ADS)

Gómez-Aguilar, J. F.

2018-03-01

In this paper, we analyze an alcoholism model which involves the impact of Twitter via Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives with constant- and variable-order. Two fractional mathematical models are considered, with and without delay. Special solutions using an iterative scheme via Laplace and Sumudu transform were obtained. We studied the uniqueness and existence of the solutions employing the fixed point postulate. The generalized model with variable-order was solved numerically via the Adams method and the Adams-Bashforth-Moulton scheme. Stability and convergence of the numerical solutions were presented in details. Numerical examples of the approximate solutions are provided to show that the numerical methods are computationally efficient. Therefore, by including both the fractional derivatives and finite time delays in the alcoholism model studied, we believe that we have established a more complete and more realistic indicator of alcoholism model and affect the spread of the drinking.

Stochastic Modelling, Analysis, and Simulations of the Solar Cycle Dynamic Process

NASA Astrophysics Data System (ADS)

Turner, Douglas C.; Ladde, Gangaram S.

2018-03-01

Analytical solutions, discretization schemes and simulation results are presented for the time delay deterministic differential equation model of the solar dynamo presented by Wilmot-Smith et al. In addition, this model is extended under stochastic Gaussian white noise parametric fluctuations. The introduction of stochastic fluctuations incorporates variables affecting the dynamo process in the solar interior, estimation error of parameters, and uncertainty of the α-effect mechanism. Simulation results are presented and analyzed to exhibit the effects of stochastic parametric volatility-dependent perturbations. The results generalize and extend the work of Hazra et al. In fact, some of these results exhibit the oscillatory dynamic behavior generated by the stochastic parametric additative perturbations in the absence of time delay. In addition, the simulation results of the modified stochastic models influence the change in behavior of the very recently developed stochastic model of Hazra et al.

Solution of differential equations by application of transformation groups

NASA Technical Reports Server (NTRS)

Driskell, C. N., Jr.; Gallaher, L. J.; Martin, R. H., Jr.

1968-01-01

Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations.

Efferent feedback can explain many hearing phenomena

NASA Astrophysics Data System (ADS)

Holmes, W. Harvey; Flax, Matthew R.

2015-12-01

The mixed mode cochlear amplifier (MMCA) model was presented at the last Mechanics of Hearing workshop [4]. The MMCA consists principally of a nonlinear feedback loop formed when an efferent-controlled outer hair cell (OHC) is combined with the cochlear mechanics and the rest of the relevant neurobiology. Essential elements of this model are efferent control of the OHC motility and a delay in the feedback to the OHC. The input to the MMCA is the passive travelling wave. In the MMCA amplification is localized where both the neural and tuned mechanical systems meet in the Organ of Corti (OoC). The simplest model based on this idea is a nonlinear delay line resonator (DLR), which is mathematically described by a nonlinear delay-differential equation (DDE). This model predicts possible Hopf bifurcations and exhibits its most interesting behaviour when operating near a bifurcation. This contribution presents some simulation results using the DLR model. These show that various observed hearing phenomena can be accounted for by this model, at least qualitatively, including compression effects, two-tone suppression and some forms of otoacoustic emissions (OAEs).

Beyond discounting: possible experimental models of impulse control.

PubMed

Monterosso, J; Ainslie, G

1999-10-01

Animal studies of impulsivity have typically used one of three models: a delay of reward procedure, a differential reinforcement for low rate responding (DRL) procedure, or an autoshaping procedure. In each of these paradigms, we argue, measurement of impulsivity is implicitly or explicitly equated with the effect delay has on the value of reward. The steepness by which delay diminishes value (the temporal discount function) is treated as an index of impulsivity. In order to provide a better analog of human impulsivity, this model needs to be expanded to include the converse of impulsivity - self-control. Through mechanisms such as committing to long range interests before the onset of temptation, or through bundling individual choices into classes of choices that are made at once, human decision-making can often look far less myopic than single trial experiments predict. For people, impulsive behavior may be more often the result of the breakdown of self-control mechanisms than of steep discount functions. Existing animal models of self-control are discussed, and future directions are suggested for psychopharmacological research.

Incorporating delayed neutrons into the point-model equations routinely used for neutron coincidence counting in nuclear safeguards

DOE PAGES

Croft, Stephen; Favalli, Andrea

2016-09-21

Here, we extend the familiar Bӧhnel point-model equations, which are routinely used to interpret neutron coincidence counting rates, by including the contribution of delayed neutrons. After developing the necessary equations we use them to show, by providing some numerical results, what the quantitative impact of neglecting delayed neutrons is across the full range of practical nuclear safeguards applications. The influence of delayed neutrons is predicted to be small for the types of deeply sub-critical assay problems which concern the nuclear safeguards community, smaller than uncertainties arising from other factors. This is most clearly demonstrated by considering the change in themore » effective (α,n)-to-spontaneous fission prompt-neutron ratio that the inclusion of delayed neutrons gives rise to. That the influence of delayed neutrons is small is fortunate, and our results justify the long standing practice of simply neglecting them in the analysis of field measurements.« less

A procedure to construct exact solutions of nonlinear fractional differential equations.

PubMed

Güner, Özkan; Cevikel, Adem C

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

Student Solution Manual for Mathematical Methods for Physics and Engineering Third Edition

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2006-03-01

Preface; 1. Preliminary algebra; 2. Preliminary calculus; 3. Complex numbers and hyperbolic functions; 4. Series and limits; 5. Partial differentiation; 6. Multiple integrals; 7. Vector algebra; 8. Matrices and vector spaces; 9. Normal modes; 10. Vector calculus; 11. Line, surface and volume integrals; 12. Fourier series; 13. Integral transforms; 14. First-order ordinary differential equations; 15. Higher-order ordinary differential equations; 16. Series solutions of ordinary differential equations; 17. Eigenfunction methods for differential equations; 18. Special functions; 19. Quantum operators; 20. Partial differential equations: general and particular; 21. Partial differential equations: separation of variables; 22. Calculus of variations; 23. Integral equations; 24. Complex variables; 25. Application of complex variables; 26. Tensors; 27. Numerical methods; 28. Group theory; 29. Representation theory; 30. Probability; 31. Statistics.

Solving Differential Equations in R: Package deSolve

EPA Science Inventory

In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...

A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

PubMed Central

Güner, Özkan; Cevikel, Adem C.

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972

A delay differential model of ENSO variability: parametric instability and the distribution of extremes

NASA Astrophysics Data System (ADS)

Zaliapin, I.; Ghil, M.; Thompson, S.

2007-12-01

We consider a Delay Differential Equation (DDE) model for El-Nino Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in the ENSO dynamics: delayed negative feedback and seasonal forcing. Descriptive and metric stability analyses of the model are performed in a complete 3D space of its physically relevant parameters. Existence of two regimes --- stable and unstable --- is reported. The domains of the regimes are separated by a sharp neutral curve in the parameter space. The detailed structure of the neutral curve become very complicated (possibly fractal), and individual trajectories within the unstable region become highly complex (possibly chaotic) as the atmosphere-ocean coupling increases. In the unstable regime, spontaneous transitions in the mean "temperature" (i.e., thermocline depth), period, and extreme annual values occur, for purely periodic, seasonal forcing. This indicates (via the continuous dependence theorem) the existence of numerous unstable solutions responsible for the complex dynamics of the system. In the stable regime, only periodic solutions are found. Our results illustrate the role of the distinct parameters of ENSO variability, such as strength of seasonal forcing vs. atmosphere ocean coupling and propagation period of oceanic waves across the Tropical Pacific. The model reproduces, among other phenomena, the Devil's bleachers (caused by period locking) documented in other ENSO models, such as nonlinear PDEs and GCMs, as well as in certain observations. We expect such behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.

An Alternative to the Stay/Switch Equation Assessed When Using a Changeover-Delay

PubMed Central

MacDonall, James S.

2015-01-01

An alternative to the generalized matching equation for understanding concurrent performances is the stay/switch model. For the stay/switch model, the important events are the contingencies and behaviors at each alternative. The current experiment compares the descriptions by two stay/switch equations, the original, empirically derived stay/switch equation and a more theoretically derived equation based on ratios of stay to switch responses matching ratios of stay to switch reinforcers. The present experiment compared descriptions by the original stay/switch equation when using and not using a changeover delay. It also compared descriptions by the more theoretical equation with and without a changeover delay. Finally, it compared descriptions of the concurrent performances by these two equations. Rats were trained in 15 conditions on identical concurrent random-interval schedules in each component of a multiple schedule. A COD operated in only one component. There were no consistent differences in the variance accounted for by each equation of concurrent performances whether or not a COD was used. The simpler equation found greater sensitivity to stay than to switch reinforcers. It also found a COD eliminated the influence of switch reinforcers. Because estimates of parameters were more meaningful when using the more theoretical stay/switch equation it is preferred. PMID:26299548

An alternative to the stay/switch equation assessed when using a changeover-delay.

PubMed

MacDonall, James S

2015-11-01

An alternative to the generalized matching equation for understanding concurrent performances is the stay/switch model. For the stay/switch model, the important events are the contingencies and behaviors at each alternative. The current experiment compares the descriptions by two stay/switch equations, the original, empirically derived stay/switch equation and a more theoretically derived equation based on ratios of stay to switch responses matching ratios of stay to switch reinforcers. The present experiment compared descriptions by the original stay/switch equation when using and not using a changeover delay. It also compared descriptions by the more theoretical equation with and without a changeover delay. Finally, it compared descriptions of the concurrent performances by these two equations. Rats were trained in 15 conditions on identical concurrent random-interval schedules in each component of a multiple schedule. A COD operated in only one component. There were no consistent differences in the variance accounted for by each equation of concurrent performances whether or not a COD was used. The simpler equation found greater sensitivity to stay than to switch reinforcers. It also found a COD eliminated the influence of switch reinforcers. Because estimates of parameters were more meaningful when using the more theoretical stay/switch equation it is preferred. Copyright © 2015 Elsevier B.V. All rights reserved.

Self-Consistent Sources Extensions of Modified Differential-Difference KP Equation

NASA Astrophysics Data System (ADS)

Gegenhasi; Li, Ya-Qian; Zhang, Duo-Duo

2018-04-01

In this paper, we investigate a modified differential-difference KP equation which is shown to have a continuum limit into the mKP equation. It is also shown that the solution of the modified differential-difference KP equation is related to the solution of the differential-difference KP equation through a Miura transformation. We first present the Grammian solution to the modified differential-difference KP equation, and then produce a coupled modified differential-difference KP system by applying the source generation procedure. The explicit N-soliton solution of the resulting coupled modified differential-difference system is expressed in compact forms by using the Grammian determinant and Casorati determinant. We also construct and solve another form of the self-consistent sources extension of the modified differential-difference KP equation, which constitutes a Bäcklund transformation for the differential-difference KP equation with self-consistent sources. Supported by the National Natural Science Foundation of China under Grant Nos. 11601247 and 11605096, the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant Nos. 2016MS0115 and 2015MS0116 and the Innovation Fund Programme of Inner Mongolia University No. 20161115

A model of partial differential equations for HIV propagation in lymph nodes

NASA Astrophysics Data System (ADS)

Marinho, E. B. S.; Bacelar, F. S.; Andrade, R. F. S.

2012-01-01

A system of partial differential equations is used to model the dissemination of the Human Immunodeficiency Virus (HIV) in CD4+T cells within lymph nodes. Besides diffusion terms, the model also includes a time-delay dependence to describe the time lag required by the immunologic system to provide defenses to new virus strains. The resulting dynamics strongly depends on the properties of the invariant sets of the model, consisting of three fixed points related to the time independent and spatial homogeneous tissue configurations in healthy and infected states. A region in the parameter space is considered, for which the time dependence of the space averaged model variables follows the clinical pattern reported for infected patients: a short scale primary infection, followed by a long latency period of almost complete recovery and third phase characterized by damped oscillations around a value with large HIV counting. Depending on the value of the diffusion coefficient, the latency time increases with respect to that one obtained for the space homogeneous version of the model. It is found that same initial conditions lead to quite different spatial patterns, which depend strongly on the latency interval.

Legendre-tau approximations for functional differential equations

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1986-01-01

The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

Legendre-Tau approximations for functional differential equations

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1983-01-01

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

Introduction to Focus Issue: Time-delay dynamics

NASA Astrophysics Data System (ADS)

Erneux, Thomas; Javaloyes, Julien; Wolfrum, Matthias; Yanchuk, Serhiy

2017-11-01

The field of dynamical systems with time delay is an active research area that connects practically all scientific disciplines including mathematics, physics, engineering, biology, neuroscience, physiology, economics, and many others. This Focus Issue brings together contributions from both experimental and theoretical groups and emphasizes a large variety of applications. In particular, lasers and optoelectronic oscillators subject to time-delayed feedbacks have been explored by several authors for their specific dynamical output, but also because they are ideal test-beds for experimental studies of delay induced phenomena. Topics include the control of cavity solitons, as light spots in spatially extended systems, new devices for chaos communication or random number generation, higher order locking phenomena between delay and laser oscillation period, and systematic bifurcation studies of mode-locked laser systems. Moreover, two original theoretical approaches are explored for the so-called Low Frequency Fluctuations, a particular chaotical regime in laser output which has attracted a lot of interest for more than 30 years. Current hot problems such as the synchronization properties of networks of delay-coupled units, novel stabilization techniques, and the large delay limit of a delay differential equation are also addressed in this special issue. In addition, analytical and numerical tools for bifurcation problems with or without noise and two reviews on concrete questions are proposed. The first review deals with the rich dynamics of simple delay climate models for El Nino Southern Oscillations, and the second review concentrates on neuromorphic photonic circuits where optical elements are used to emulate spiking neurons. Finally, two interesting biological problems are considered in this Focus Issue, namely, multi-strain epidemic models and the interaction of glucose and insulin for more effective treatment.

A gradual update method for simulating the steady-state solution of stiff differential equations in metabolic circuits.

PubMed

Shiraishi, Emi; Maeda, Kazuhiro; Kurata, Hiroyuki

2009-02-01

Numerical simulation of differential equation systems plays a major role in the understanding of how metabolic network models generate particular cellular functions. On the other hand, the classical and technical problems for stiff differential equations still remain to be solved, while many elegant algorithms have been presented. To relax the stiffness problem, we propose new practical methods: the gradual update of differential-algebraic equations based on gradual application of the steady-state approximation to stiff differential equations, and the gradual update of the initial values in differential-algebraic equations. These empirical methods show a high efficiency for simulating the steady-state solutions for the stiff differential equations that existing solvers alone cannot solve. They are effective in extending the applicability of dynamic simulation to biochemical network models.

Probabilistic delay differential equation modeling of event-related potentials.

PubMed

Ostwald, Dirk; Starke, Ludger

2016-08-01

"Dynamic causal models" (DCMs) are a promising approach in the analysis of functional neuroimaging data due to their biophysical interpretability and their consolidation of functional-segregative and functional-integrative propositions. In this theoretical note we are concerned with the DCM framework for electroencephalographically recorded event-related potentials (ERP-DCM). Intuitively, ERP-DCM combines deterministic dynamical neural mass models with dipole-based EEG forward models to describe the event-related scalp potential time-series over the entire electrode space. Since its inception, ERP-DCM has been successfully employed to capture the neural underpinnings of a wide range of neurocognitive phenomena. However, in spite of its empirical popularity, the technical literature on ERP-DCM remains somewhat patchy. A number of previous communications have detailed certain aspects of the approach, but no unified and coherent documentation exists. With this technical note, we aim to close this gap and to increase the technical accessibility of ERP-DCM. Specifically, this note makes the following novel contributions: firstly, we provide a unified and coherent review of the mathematical machinery of the latent and forward models constituting ERP-DCM by formulating the approach as a probabilistic latent delay differential equation model. Secondly, we emphasize the probabilistic nature of the model and its variational Bayesian inversion scheme by explicitly deriving the variational free energy function in terms of both the likelihood expectation and variance parameters. Thirdly, we detail and validate the estimation of the model with a special focus on the explicit form of the variational free energy function and introduce a conventional nonlinear optimization scheme for its maximization. Finally, we identify and discuss a number of computational issues which may be addressed in the future development of the approach. Copyright © 2016 Elsevier Inc. All rights reserved.

The existence of solutions of q-difference-differential equations.

PubMed

Wang, Xin-Li; Wang, Hua; Xu, Hong-Yan

2016-01-01

By using the Nevanlinna theory of value distribution, we investigate the existence of solutions of some types of non-linear q-difference differential equations. In particular, we generalize the Rellich-Wittich-type theorem and Malmquist-type theorem about differential equations to the case of q-difference differential equations (system).

Reaction-diffusion systems in natural sciences and new technology transfer

NASA Astrophysics Data System (ADS)

Keller, André A.

2012-12-01

Diffusion mechanisms in natural sciences and innovation management involve partial differential equations (PDEs). This is due to their spatio-temporal dimensions. Functional semi-discretized PDEs (with lattice spatial structures or time delays) may be even more adapted to real world problems. In the modeling process, PDEs can also formalize behaviors, such as the logistic growth of populations with migration, and the adopters’ dynamics of new products in innovation models. In biology, these events are related to variations in the environment, population densities and overcrowding, migration and spreading of humans, animals, plants and other cells and organisms. In chemical reactions, molecules of different species interact locally and diffuse. In the management of new technologies, the diffusion processes of innovations in the marketplace (e.g., the mobile phone) are a major subject. These innovation diffusion models refer mainly to epidemic models. This contribution introduces that modeling process by using PDEs and reviews the essential features of the dynamics and control in biological, chemical and new technology transfer. This paper is essentially user-oriented with basic nonlinear evolution equations, delay PDEs, several analytical and numerical methods for solving, different solutions, and with the use of mathematical packages, notebooks and codes. The computations are carried out by using the software Wolfram Mathematica®7, and C++ codes.

Angular-Rate Estimation Using Delayed Quaternion Measurements

NASA Technical Reports Server (NTRS)

Azor, R.; Bar-Itzhack, I. Y.; Harman, R. R.

1999-01-01

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared one that uses differentiated quaternion measurements to yield coarse rate measurements, which are then fed into two different estimators. In the other approach the raw quaternion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear part of the rotas rotational dynamics equation of a body into a product of an angular-rate dependent matrix and the angular-rate vector itself. This non unique decomposition, enables the treatment of the nonlinear spacecraft (SC) dynamics model as a linear one and, thus, the application of a PseudoLinear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the gain matrix and thus eliminates the need to compute recursively the filter covariance matrix. The replacement of the rotational dynamics by a simple Markov model is also examined. In this paper special consideration is given to the problem of delayed quaternion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results are presented.

A statistical rain attenuation prediction model with application to the advanced communication technology satellite project. Part 2: Theoretical development of a dynamic model and application to rain fade durations and tolerable control delays for fade countermeasures

NASA Technical Reports Server (NTRS)

Manning, Robert M.

1987-01-01

A dynamic rain attenuation prediction model is developed for use in obtaining the temporal characteristics, on time scales of minutes or hours, of satellite communication link availability. Analagous to the associated static rain attenuation model, which yields yearly attenuation predictions, this dynamic model is applicable at any location in the world that is characterized by the static rain attenuation statistics peculiar to the geometry of the satellite link and the rain statistics of the location. Such statistics are calculated by employing the formalism of Part I of this report. In fact, the dynamic model presented here is an extension of the static model and reduces to the static model in the appropriate limit. By assuming that rain attenuation is dynamically described by a first-order stochastic differential equation in time and that this random attenuation process is a Markov process, an expression for the associated transition probability is obtained by solving the related forward Kolmogorov equation. This transition probability is then used to obtain such temporal rain attenuation statistics as attenuation durations and allowable attenuation margins versus control system delay.

Minimum-variance Brownian motion control of an optically trapped probe.

PubMed

Huang, Yanan; Zhang, Zhipeng; Menq, Chia-Hsiang

2009-10-20

This paper presents a theoretical and experimental investigation of the Brownian motion control of an optically trapped probe. The Langevin equation is employed to describe the motion of the probe experiencing random thermal force and optical trapping force. Since active feedback control is applied to suppress the probe's Brownian motion, actuator dynamics and measurement delay are included in the equation. The equation of motion is simplified to a first-order linear differential equation and transformed to a discrete model for the purpose of controller design and data analysis. The derived model is experimentally verified by comparing the model prediction to the measured response of a 1.87 microm trapped probe subject to proportional control. It is then employed to design the optimal controller that minimizes the variance of the probe's Brownian motion. Theoretical analysis is derived to evaluate the control performance of a specific optical trap. Both experiment and simulation are used to validate the design as well as theoretical analysis, and to illustrate the performance envelope of the active control. Moreover, adaptive minimum variance control is implemented to maintain the optimal performance in the case in which the system is time varying when operating the actively controlled optical trap in a complex environment.

Role of the noise on the transient dynamics of an ecosystem of interacting species

NASA Astrophysics Data System (ADS)

Spagnolo, B.; La Barbera, A.

2002-11-01

We analyze the transient dynamics of an ecosystem described by generalized Lotka-Volterra equations in the presence of a multiplicative noise and a random interaction parameter between the species. We consider specifically three cases: (i) two competing species, (ii) three interacting species (one predator-two preys), (iii) n-interacting species. The interaction parameter in case (i) is a stochastic process which obeys a stochastic differential equation. We find noise delayed extinction of one of two species, which is akin to the noise-enhanced stability phenomenon. Other two noise-induced effects found are temporal oscillations and spatial patterns of the two competing species. In case (ii) the noise induces correlated spatial patterns of the predator and of the two preys concentrations. Finally, in case (iii) we find the asymptotic behavior of the time average of the ith population when the ecosystem is composed of a great number of interacting species.

The convergence of the order sequence and the solution function sequence on fractional partial differential equation

NASA Astrophysics Data System (ADS)

Rusyaman, E.; Parmikanti, K.; Chaerani, D.; Asefan; Irianingsih, I.

2018-03-01

One of the application of fractional ordinary differential equation is related to the viscoelasticity, i.e., a correlation between the viscosity of fluids and the elasticity of solids. If the solution function develops into function with two or more variables, then its differential equation must be changed into fractional partial differential equation. As the preliminary study for two variables viscoelasticity problem, this paper discusses about convergence analysis of function sequence which is the solution of the homogenous fractional partial differential equation. The method used to solve the problem is Homotopy Analysis Method. The results show that if given two real number sequences (αn) and (βn) which converge to α and β respectively, then the solution function sequences of fractional partial differential equation with order (αn, βn) will also converge to the solution function of fractional partial differential equation with order (α, β).

Informed Conjecturing of Solutions for Differential Equations in a Modeling Context

ERIC Educational Resources Information Center

Winkel, Brian

2015-01-01

We examine two differential equations. (i) first-order exponential growth or decay; and (ii) second order, linear, constant coefficient differential equations, and show the advantage of learning differential equations in a modeling context for informed conjectures of their solution. We follow with a discussion of the complete analysis afforded by…

The Trade-Off Mechanism in Mammalian Circadian Clock Model with Two Time Delays

NASA Astrophysics Data System (ADS)

Yan, Jie; Kang, Xiaxia; Yang, Ling

Circadian clock is an autonomous oscillator which orchestrates the daily rhythms of physiology and behaviors. This study is devoted to explore how a positive feedback loop affects the dynamics of mammalian circadian clock. We simplify an experimentally validated mathematical model in our previous work, to a nonlinear differential equation with two time delays. This simplified mathematical model incorporates the pacemaker of mammalian circadian clock, a negative primary feedback loop, and a critical positive auxiliary feedback loop, Rev-erbα/Cry1 loop. We perform analytical studies of the system. Delay-dependent conditions for the asymptotic stability of the nontrivial positive steady state of the model are investigated. We also prove the existence of Hopf bifurcation, which leads to self-sustained oscillation of mammalian circadian clock. Our theoretical analyses show that the oscillatory regime is reduced upon the participation of the delayed positive auxiliary loop. However, further simulations reveal that the auxiliary loop can enable the circadian clock gain widely adjustable amplitudes and robust period. Thus, the positive auxiliary feedback loop may provide a trade-off mechanism, to use the small loss in the robustness of oscillation in exchange for adaptable flexibility in mammalian circadian clock. The results obtained from the model may gain new insights into the dynamics of biological oscillators with interlocked feedback loops.

Schwarz maps of algebraic linear ordinary differential equations

NASA Astrophysics Data System (ADS)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

On the Inclusion of Difference Equation Problems and Z Transform Methods in Sophomore Differential Equation Classes

ERIC Educational Resources Information Center

Savoye, Philippe

2009-01-01

In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.

Minimizing Secular J2 Perturbation Effects on Satellite Formations

DTIC Science & Technology

2008-03-01

linear set of differential equations describing the relative motion was established by Hill as well as Clohessy and Wiltshire , with a slightly... Wiltshire (CW) equations, and Hill- Clohessy - Wiltshire (HCW) equations. In the simplest form these differential equations can be expressed as: 2 2 2 3 2...different orientation. Because these equations are much alike, the differential equations established are referred to as Hill’s equations, Clohessy

A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type,

DTIC Science & Technology

NONLINEAR DIFFERENTIAL EQUATIONS, INTEGRATION), (*PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS), BANACH SPACE , MAPPING (TRANSFORMATIONS), SET THEORY, TOPOLOGY, ITERATIONS, STABILITY, THEOREMS

Dynamic characteristics of a variable-mass flexible missile

NASA Technical Reports Server (NTRS)

Meirovitch, L.; Bankovskis, J.

1970-01-01

The general motion of a variable mass flexible missile with internal flow and aerodynamic forces is considered. The resulting formulation comprises six ordinary differential equations for rigid body motion and three partial differential equations for elastic motion. The simultaneous differential equations are nonlinear and possess time-dependent coefficients. The differential equations are solved by a semi-analytical method leading to a set of purely ordinary differential equations which are then solved numerically. A computer program was developed for the numerical solution and results are presented for a given set of initial conditions.

Kalman filter estimation of human pilot-model parameters

NASA Technical Reports Server (NTRS)

Schiess, J. R.; Roland, V. R.

1975-01-01

The parameters of a human pilot-model transfer function are estimated by applying the extended Kalman filter to the corresponding retarded differential-difference equations in the time domain. Use of computer-generated data indicates that most of the parameters, including the implicit time delay, may be reasonably estimated in this way. When applied to two sets of experimental data obtained from a closed-loop tracking task performed by a human, the Kalman filter generated diverging residuals for one of the measurement types, apparently because of model assumption errors. Application of a modified adaptive technique was found to overcome the divergence and to produce reasonable estimates of most of the parameters.

Fully unsteady subsonic and supersonic potential aerodynamics for complex aircraft configurations for flutter applications

NASA Technical Reports Server (NTRS)

Tseng, K.; Morino, L.

1975-01-01

A general theory for study, oscillatory or fully unsteady potential compressible aerodynamics around complex configurations is presented. Using the finite-element method to discretize the space problem, one obtains a set of differential-delay equations in time relating the potential to its normal derivative which is expressed in terms of the generalized coordinates of the structure. For oscillatory flow, the motion consists of sinusoidal oscillations around a steady, subsonic or supersonic flow. For fully unsteady flow, the motion is assumed to consist of constant subsonic or supersonic speed for time t or = 0 and of small perturbations around the steady state for time t 0.

Stability of equations with a distributed delay, monotone production and nonlinear mortality

NASA Astrophysics Data System (ADS)

Berezansky, Leonid; Braverman, Elena

2013-10-01

We consider population dynamics models dN/dt = f(N(tτ)) - d(N(t)) with an increasing fecundity function f and any mortality function d which can be quadratic, as in the logistic equation, or have a different form provided that the equation has at most one positive equilibrium. Here the delay in the production term can be distributed and unbounded. It is demonstrated that the positive equilibrium is globally attractive if it exists, otherwise all positive solutions tend to zero. Moreover, we demonstrate that solutions of the equation are intrinsically non-oscillatory: once the initial function is less/greater than the equilibrium K > 0, so is the solution for any positive time value. The assumptions on f, d and the delay are rather nonrestrictive, and several examples demonstrate that none of them can be omitted.

FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations

NASA Astrophysics Data System (ADS)

Ibragimov, N. H.; Torrisi, M.; Tracinà, R.

2010-11-01

In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

Derivation of kinetic equations from non-Wiener stochastic differential equations

NASA Astrophysics Data System (ADS)

Basharov, A. M.

2013-12-01

Kinetic differential-difference equations containing terms with fractional derivatives and describing α -stable Levy processes with 0 < α < 1 have been derived in a unified manner in terms of one-dimensional stochastic differential equations controlled merely by the Poisson processes.

A Tribute to J. C. Sprott

NASA Astrophysics Data System (ADS)

Nazarimehr, Fahimeh; Jafari, Sajad; Chen, Guanrong; Kapitaniak, Tomasz; Kuznetsov, Nikolay V.; Leonov, Gennady A.; Li, Chunbiao; Wei, Zhouchao

2017-12-01

In honor of his 75th birthday, we review the prominent works of Professor Julien Clinton Sprott in chaos and nonlinear dynamics. We categorize his works into three important groups. The first and most important group is identifying new dynamical systems with special properties. He has proposed different chaotic maps, flows, complex variable systems, nonautonomous systems, partial differential equations, fractional-order systems, delay differential systems, spatiotemporal systems, artificial neural networks, and chaotic electrical circuits. He has also studied dynamical properties of complex systems such as bifurcations and basins of attraction. He has done work on generating fractal art. He has examined models of real-world systems that exhibit chaos. The second group of his works comprise control and synchronization of chaos. Finally, the third group is extracting dynamical properties of systems using time-series analysis. This paper highlights the impact of Sprott’s work on the promotion of nonlinear dynamics.

A model with competition between the cell lines in leukemia under treatment

DOE Office of Scientific and Technical Information (OSTI.GOV)

Halanay, A.; Cândea, D.; Rădulescu, R.

2014-12-10

The evolution of leukemia is modeled with a delay differential equation model of four cell populations: two populations (healthy and leukemic) ) of stem-like cells involving a larger category consisting of proliferating stem and progenitor cells with self-renew capacity and two populations (healthy and leukemic) of mature cells, considering the competition of healthy vs. leukemic cell populations and three types of division that a stem-like cell can exhibit: self-renew, asymmetric division and differentiation. In the model it is assumed that the treatment acts on the proliferation rate of the leukemic stem cells and on the apoptosis of stem and maturemore » cells. The emphasis in this model is on establishing relevant parameters for chronic and acute manifestations of leukemia. Stability of equilibria is investigated and sufficient conditions for local asymptotic stability will be given using a Lyapunov-Krasovskii functional.« less

Modular Expression Language for Ordinary Differential Equation Editing

DOE Office of Scientific and Technical Information (OSTI.GOV)

Blake, Robert C.

MELODEEis a system for describing systems of initial value problem ordinary differential equations, and a compiler for the language that produces optimized code to integrate the differential equations. Features include rational polynomial approximation for expensive functions and automatic differentiation for symbolic jacobians

Application of the Sumudu Transform to Discrete Dynamic Systems

ERIC Educational Resources Information Center

Asiru, Muniru Aderemi

2003-01-01

The Sumudu transform is an integral transform introduced to solve differential equations and control engineering problems. The transform possesses many interesting properties that make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and…

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (3).

PubMed

Murase, Kenya

2016-01-01

In this issue, simultaneous differential equations were introduced. These differential equations are often used in the field of medical physics. The methods for solving them were also introduced, which include Laplace transform and matrix methods. Some examples were also introduced, in which Laplace transform and matrix methods were applied to solving simultaneous differential equations derived from a three-compartment kinetic model for analyzing the glucose metabolism in tissues and Bloch equations for describing the behavior of the macroscopic magnetization in magnetic resonance imaging.In the next (final) issue, partial differential equations and various methods for solving them will be introduced together with some examples in medical physics.

On implicit abstract neutral nonlinear differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br; O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie

2016-04-15

In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.

FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

NASA Astrophysics Data System (ADS)

Man, Yiu-Kwong

2010-10-01

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

Solving Differential Equations Analytically. Elementary Differential Equations. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Unit 335.

ERIC Educational Resources Information Center

Goldston, J. W.

This unit introduces analytic solutions of ordinary differential equations. The objective is to enable the student to decide whether a given function solves a given differential equation. Examples of problems from biology and chemistry are covered. Problem sets, quizzes, and a model exam are included, and answers to all items are provided. The…

Approximation of Quantum Stochastic Differential Equations for Input-Output Model Reduction

DTIC Science & Technology

2016-02-25

Approximation of Quantum Stochastic Differential Equations for Input-Output Model Reduction We have completed a short program of theoretical research...on dimensional reduction and approximation of models based on quantum stochastic differential equations. Our primary results lie in the area of...2211 quantum probability, quantum stochastic differential equations REPORT DOCUMENTATION PAGE 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 10. SPONSOR

Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models

NASA Astrophysics Data System (ADS)

Pukite, P. R.

2017-12-01

By solving Laplace's tidal equations along the equatorial Pacific thermocline, assuming a delayed-differential effective gravity forcing due to a combined lunar+solar (lunisolar) stimulus, we are able to precisely match ENSO periodic variations over wide intervals. The underlying pattern is difficult to decode by conventional means such as spectral analysis, which is why it has remained hidden for so long, despite the excellent agreement in the time-domain. What occurs is that a non-linear seasonal modulation with monthly and fortnightly lunar impulses along with a biennially-aligned "see-saw" is enough to cause a physical aliasing and thus multiple folding in the frequency spectrum. So, instead of a conventional spectral tidal decomposition, we opted for a time-domain cross-validating approach to calibrate the amplitude and phasing of the lunisolar cycles. As the lunar forcing consists of three fundamental periods (draconic, anomalistic, synodic), we used the measured Earth's length-of-day (LOD) decomposed and resolved at a monthly time-scale [1] to align the amplitude and phase precisely. Even slight variations from the known values of the long-period tides will degrade the fit, so a high-resolution calibration is possible. Moreover, a narrow training segment from 1880-1920 using NINO34/SOI data is adequate to extrapolate the cycles of the past 100 years (see attached figure). To further understand the biennial impact of a yearly differential-delay, we were able to also decompose using difference equations the historical sea-level-height readings at Sydney harbor to clearly expose the ENSO behavior. Finally, the ENSO lunisolar model was validated by back-extrapolating to Unified ENSO coral proxy (UEP) records dating to 1650. The quasi-biennial oscillation (QBO) behavior of equatorial stratospheric winds derives following a similar pattern to ENSO via the tidal equations, but with an emphasis on draconic forcing. This improvement in ENSO and QBO understanding has implications for vastly simplifying global climate models due to the straightforward application of a well-known and well-calibrated forcing. [1] Na, Sung-Ho, et al. "Characteristics of Perturbations in Recent Length of Day and Polar Motion." Journal of Astronomy and Space Sciences 30 (2013): 33-41.

Numerical modelling of multimode fibre-optic communication lines

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sidelnikov, O S; Fedoruk, M P; Sygletos, S

The results of numerical modelling of nonlinear propagation of an optical signal in multimode fibres with a small differential group delay are presented. It is found that the dependence of the error vector magnitude (EVM) on the differential group delay can be reduced by increasing the number of ADC samples per symbol in the numerical implementation of the differential group delay compensation algorithm in the receiver. The possibility of using multimode fibres with a small differential group delay for data transmission in modern digital communication systems is demonstrated. It is shown that with increasing number of modes the strong couplingmore » regime provides a lower EVM level than the weak coupling one. (fibre-optic communication lines)« less

Fault Tolerant Optimal Control.

DTIC Science & Technology

1982-08-01

subsystem is modelled by deterministic or stochastic finite-dimensional vector differential or difference equations. The parameters of these equations...is no partial differential equation that must be solved. Thus we can sidestep the inability to solve the Bellman equation for control problems with x...transition models and cost functionals can be reduced to the search for solutions of nonlinear partial differential equations using ’verification

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Angstmann, C.N.; Donnelly, I.C.; Henry, B.I., E-mail: B.Henry@unsw.edu.au

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also showmore » that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.« less

An accessible four-dimensional treatment of Maxwell's equations in terms of differential forms

NASA Astrophysics Data System (ADS)

Sá, Lucas

2017-03-01

Maxwell’s equations are derived in terms of differential forms in the four-dimensional Minkowski representation, starting from the three-dimensional vector calculus differential version of these equations. Introducing all the mathematical and physical concepts needed (including the tool of differential forms), using only knowledge of elementary vector calculus and the local vector version of Maxwell’s equations, the equations are reduced to a simple and elegant set of two equations for a unified quantity, the electromagnetic field. The treatment should be accessible for students taking a first course on electromagnetism.

Periodic synchronization control of discontinuous delayed networks by using extended Filippov-framework.

PubMed

Cai, Zuowei; Huang, Lihong; Guo, Zhenyuan; Zhang, Lingling; Wan, Xuting

2015-08-01

This paper is concerned with the periodic synchronization problem for a general class of delayed neural networks (DNNs) with discontinuous neuron activation. One of the purposes is to analyze the problem of periodic orbits. To do so, we introduce new tools including inequality techniques and Kakutani's fixed point theorem of set-valued maps to derive the existence of periodic solution. Another purpose is to design a switching state-feedback control for realizing global exponential synchronization of the drive-response network system with periodic coefficients. Unlike the previous works on periodic synchronization of neural network, both the neuron activations and controllers in this paper are allowed to be discontinuous. Moreover, owing to the occurrence of delays in neuron signal, the neural network model is described by the functional differential equation. So we introduce extended Filippov-framework to deal with the basic issues of solutions for discontinuous DNNs. Finally, two examples and simulation experiments are given to illustrate the proposed method and main results which have an important instructional significance in the design of periodic synchronized DNNs circuits involving discontinuous or switching factors. Copyright © 2015 Elsevier Ltd. All rights reserved.

Linear stability and nonlinear analyses of traffic waves for the general nonlinear car-following model with multi-time delays

NASA Astrophysics Data System (ADS)

Sun, Dihua; Chen, Dong; Zhao, Min; Liu, Weining; Zheng, Linjiang

2018-07-01

In this paper, the general nonlinear car-following model with multi-time delays is investigated in order to describe the reactions of vehicle to driving behavior. Platoon stability and string stability criteria are obtained for the general nonlinear car-following model. Burgers equation and Korteweg de Vries (KdV) equation and their solitary wave solutions are derived adopting the reductive perturbation method. We investigate the properties of typical optimal velocity model using both analytic and numerical methods, which estimates the impact of delays about the evolution of traffic congestion. The numerical results show that time delays in sensing relative movement is more sensitive to the stability of traffic flow than time delays in sensing host motion.

Chaotic attractors in tumor growth and decay: a differential equation model.

PubMed

Harney, Michael; Yim, Wen-sau

2015-01-01

Tumorigenesis can be modeled as a system of chaotic nonlinear differential equations. A simulation of the system is realized by converting the differential equations to difference equations. The results of the simulation show that an increase in glucose in the presence of low oxygen levels decreases tumor growth.

The Algorithm Theoretical Basis Document for the Atmospheric Delay Correction to GLAS Laser Altimeter Ranges. Volume 8

NASA Technical Reports Server (NTRS)

Herring, Thomas A.; Quinn, Katherine J.

2012-01-01

NASA s Ice, Cloud, and Land Elevation Satellite (ICESat) mission will be launched late 2001. It s primary instrument is the Geoscience Laser Altimeter System (GLAS) instrument. The main purpose of this instrument is to measure elevation changes of the Greenland and Antarctic icesheets. To accurately measure the ranges it is necessary to correct for the atmospheric delay of the laser pulses. The atmospheric delay depends on the integral of the refractive index along the path that the laser pulse travels through the atmosphere. The refractive index of air at optical wavelengths is a function of density and molecular composition. For ray paths near zenith and closed form equations for the refractivity, the atmospheric delay can be shown to be directly related to surface pressure and total column precipitable water vapor. For ray paths off zenith a mapping function relates the delay to the zenith delay. The closed form equations for refractivity recommended by the International Union of Geodesy and Geophysics (IUGG) are optimized for ground based geodesy techniques and in the next section we will consider whether these equations are suitable for satellite laser altimetry.

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

NASA Astrophysics Data System (ADS)

Da Rocha, R.; Capelas Oliveira, E.

2009-01-01

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Börner-Dürr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.

Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms.

PubMed

Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan

2017-04-07

In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.

A new approach to Catalan numbers using differential equations

NASA Astrophysics Data System (ADS)

Kim, D. S.; Kim, T.

2017-10-01

In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in a certain sense. From these differential equations, we obtain some new and explicit identities for Catalan and higher-order Catalan numbers. In addition, by other means than differential equations, we also derive some interesting identities involving Catalan numbers which are of arithmetic and combinatorial nature.

A Malaria Transmission Model with Temperature-Dependent Incubation Period.

PubMed

Wang, Xiunan; Zhao, Xiao-Qiang

2017-05-01

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio [Formula: see text] and establish a threshold type result on the global dynamics in terms of [Formula: see text], that is, the unique disease-free periodic solution is globally asymptotically stable if [Formula: see text]; and the model system admits a unique positive periodic solution which is globally asymptotically stable if [Formula: see text]. Numerically, we parameterize the model with data from Maputo Province, Mozambique, and simulate the long-term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may underestimate the basic reproduction ratio.

Dynamic Analysis of a Reaction-Diffusion Rumor Propagation Model

NASA Astrophysics Data System (ADS)

Zhao, Hongyong; Zhu, Linhe

2016-06-01

The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. Rumor propagation in social networks has brought new challenges to network security and social stability. This paper, based on partial differential equations (PDEs), proposes a new SIS rumor propagation model by considering the effect of the communication between the different rumor infected users on rumor propagation. The stabilities of a nonrumor equilibrium point and a rumor-spreading equilibrium point are discussed by linearization technique and the upper and lower solutions method, and the existence of a traveling wave solution is established by the cross-iteration scheme accompanied by the technique of upper and lower solutions and Schauder’s fixed point theorem. Furthermore, we add the time delay to rumor propagation and deduce the conditions of Hopf bifurcation and stability switches for the rumor-spreading equilibrium point by taking the time delay as the bifurcation parameter. Finally, numerical simulations are performed to illustrate the theoretical results.

Charge modeling of ionic polymer-metal composites for dynamic curvature sensing

NASA Astrophysics Data System (ADS)

Bahramzadeh, Yousef; Shahinpoor, Mohsen

2011-04-01

A curvature sensor based on Ionic Polymer-Metal Composite (IPMC) is proposed and characterized for sensing of curvature variation in structures such as inflatable space structures in which using low power and flexible curvature sensor is of high importance for dynamic monitoring of shape at desired points. The linearity of output signal of sensor for calibration, effect of deflection rate at low frequencies and the phase delay between the output signal and the input deformation of IPMC curvature sensor is investigated. An analytical chemo-electro-mechanical model for charge dynamic of IPMC sensor is presented based on Nernst-Planck partial differential equation which can be used to explain the phenomena observed in experiments. The rate dependency of output signal and phase delay between the applied deformation and sensor signal is studied using the proposed model. The model provides a background for predicting the general characteristics of IPMC sensor. It is shown that IPMC sensor exhibits good linearity, sensitivity, and repeatability for dynamic curvature sensing of inflatable structures.

Variations in the Solution of Linear First-Order Differential Equations. Classroom Notes

ERIC Educational Resources Information Center

Seaman, Brian; Osler, Thomas J.

2004-01-01

A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx)+p(x)v=q(x) by means of a substitution v=f(y). The student then creates a table of the new equations and…

Use of artificial bee colonies algorithm as numerical approximation of differential equations solution

NASA Astrophysics Data System (ADS)

Fikri, Fariz Fahmi; Nuraini, Nuning

2018-03-01

The differential equation is one of the branches in mathematics which is closely related to human life problems. Some problems that occur in our life can be modeled into differential equations as well as systems of differential equations such as the Lotka-Volterra model and SIR model. Therefore, solving a problem of differential equations is very important. Some differential equations are difficult to solve, so numerical methods are needed to solve that problems. Some numerical methods for solving differential equations that have been widely used are Euler Method, Heun Method, Runge-Kutta and others. However, some of these methods still have some restrictions that cause the method cannot be used to solve more complex problems such as an evaluation interval that we cannot change freely. New methods are needed to improve that problems. One of the method that can be used is the artificial bees colony algorithm. This algorithm is one of metaheuristic algorithm method, which can come out from local search space and do exploration in solution search space so that will get better solution than other method.

Delay-induced wave instabilities in single-species reaction-diffusion systems

NASA Astrophysics Data System (ADS)

Otto, Andereas; Wang, Jian; Radons, Günter

2017-11-01

The Turing (wave) instability is only possible in reaction-diffusion systems with more than one (two) components. Motivated by the fact that a time delay increases the dimension of a system, we investigate the presence of diffusion-driven instabilities in single-species reaction-diffusion systems with delay. The stability of arbitrary one-component systems with a single discrete delay, with distributed delay, or with a variable delay is systematically analyzed. We show that a wave instability can appear from an equilibrium of single-species reaction-diffusion systems with fluctuating or distributed delay, which is not possible in similar systems with constant discrete delay or without delay. More precisely, we show by basic analytic arguments and by numerical simulations that fast asymmetric delay fluctuations or asymmetrically distributed delays can lead to wave instabilities in these systems. Examples, for the resulting traveling waves are shown for a Fisher-KPP equation with distributed delay in the reaction term. In addition, we have studied diffusion-induced instabilities from homogeneous periodic orbits in the same systems with variable delay, where the homogeneous periodic orbits are attracting resonant periodic solutions of the system without diffusion, i.e., periodic orbits of the Hutchinson equation with time-varying delay. If diffusion is introduced, standing waves can emerge whose temporal period is equal to the period of the variable delay.

About Global Stable of Solutions of Logistic Equation with Delay

NASA Astrophysics Data System (ADS)

Kaschenko, S. A.; Loginov, D. O.

2017-12-01

The article is devoted to the definition of all the arguments for which all positive solutions of logistic equation with delay tend to zero for t → ∞. The authors have proved the acquainted Wright’s conjecture on evaluation of a multitude of such arguments. An approach that enables subsequent refinement of this evaluation has been developed.

Dynamics of temporally localized states in passively mode-locked semiconductor lasers

NASA Astrophysics Data System (ADS)

Schelte, C.; Javaloyes, J.; Gurevich, S. V.

2018-05-01

We study the emergence and the stability of temporally localized structures in the output of a semiconductor laser passively mode locked by a saturable absorber in the long-cavity regime. For large yet realistic values of the linewidth enhancement factor, we disclose the existence of secondary dynamical instabilities where the pulses develop regular and subsequent irregular temporal oscillations. By a detailed bifurcation analysis we show that additional solution branches that consist of multipulse (molecules) solutions exist. We demonstrate that the various solution curves for the single and multipeak pulses can splice and intersect each other via transcritical bifurcations, leading to a complex web of solutions. Our analysis is based on a generic model of mode locking that consists of a time-delayed dynamical system, but also on a much more numerically efficient, yet approximate, partial differential equation. We compare the results of the bifurcation analysis of both models in order to assess up to which point the two approaches are equivalent. We conclude our analysis by the study of the influence of group velocity dispersion, which is only possible in the framework of the partial differential equation model, and we show that it may have a profound impact on the dynamics of the localized states.

From differential to difference equations for first order ODEs

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Walker, Kevin P.

1991-01-01

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Alleviation of whirl-flutter on a joined-wing tilt-rotor aircraft configuration using active controls

NASA Technical Reports Server (NTRS)

Vanaken, Johannes M.

1991-01-01

The feasibility of using active controls to delay the onset of whirl-flutter on a joined-wing tilt rotor aircraft was investigated. The CAMRAD/JA code was used to obtain a set of linear differential equations which describe the motion of the joined-wing tilt-rotor aircraft. The hub motions due to wing/body motion is a standard input to CAMRAD/JA and were obtained from a structural dynamics model of a representative joined-wing tilt-rotor aircraft. The CAMRAD/JA output, consisting of the open-loop system matrices, and the airframe free vibration motion were input to a separate program which performed the closed-loop, active control calculations. An eigenvalue analysis was performed to determine the flutter stability of both open- and closed-loop systems. Sensor models, based upon the feedback of pure state variables and based upon hub-mounted sensors, providing physically measurable accelerations, were evaluated. It was shown that the onset of tilt-rotor whirl-flutter could be delayed from 240 to above 270 knots by feeding back vertical and span-wise accelerations, measured at the rotor hub, to the longitudinal cyclic pitch. Time response calculations at a 270-knot cruise condition showed an active cyclic pitch control level of 0.009 deg, which equates to a very acceptable 9 pound active-control force applied at the rotor hub.

Differentiation of original and regenerated skeletal muscle fibres in mdx dystrophic muscles.

PubMed

Earnshaw, John C; Kyprianou, Phillip; Krishan, Kewal; Dhoot, Gurtej K

2002-07-01

The differentiation of both original muscle fibres and the regenerated muscle fibres following necrosis in mdx muscles was investigated using immunoblotting and immunocytochemical procedures. Before the onset of necrosis, postnatal skeletal muscles in mdx mouse differentiated well with only a slight delay in differentiation indicated by the level of developmental isoforms of troponin T. Prior to the onset of apparent myopathic change, both fast and slow skeletal muscle fibre types in mdx leg muscles also differentiated well when investigated by analysis of specific myosin heavy chain expression pattern. While the original muscle fibres in mdx leg muscles developed well, the differentiation of regenerated myotubes into both slow and distinct fast muscle fibre types, however, was markedly delayed or inhibited as indicated by several clusters of homogeneously staining fibres even at 14 weeks of age. The number of slow myosin heavy chain-positive myotubes amongst the regenerated muscle clusters was quite small even in soleus. This study thus established that while muscle fibres initially develop normally with only a slight delay in the differentiation process, the differentiation of regenerated myotubes in mdx muscles is markedly compromised and consequently delayed.

Distributed delays in a hybrid model of tumor-immune system interplay.

PubMed

Caravagna, Giulio; Graudenzi, Alex; d'Onofrio, Alberto

2013-02-01

A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect. We here present a hybrid model with a generic delay kernel accounting that, due to many complex phenomena such as chemical transportation and cellular differentiation, the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model with two well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process. Via simulations and parametric sensitivity analysis techniques we (i) relate tumor mass growth with the two kernels, we (ii) measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and (iii) we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication.

Nonlinear differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis ismore » on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.« less

The performance of differential VLBI delay during interplanetary cruise

NASA Technical Reports Server (NTRS)

Moultrie, B.; Wolff, P. J.; Taylor, T. H.

1984-01-01

Project Voyager radio metric data are used to evaluate the orbit determination abilities of several data strategies during spacecraft interplanetary cruise. Benchmark performance is established with an operational data strategy of conventional coherent doppler, coherent range, and explicitly differenced range data from two intercontinental baselines to ameliorate the low declination singularity of the doppler data. Employing a Voyager operations trajectory as a reference, the performance of the operational data strategy is compared to the performances of data strategies using differential VLBI delay data (spacecraft delay minus quasar delay) in combinations with the aforementioned conventional data types. The comparison of strategy performances indicates that high accuracy cruise orbit determination can be achieved with a data strategy employing differential VLBI delay data, where the quantity of coherent radio metric data has been greatly reduced.

Differential equations driven by rough paths with jumps

NASA Astrophysics Data System (ADS)

Friz, Peter K.; Zhang, Huilin

2018-05-01

We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.

Multiple Scenarios of Transition to Chaos in the Alternative Splicing Model

NASA Astrophysics Data System (ADS)

Kogai, Vladislav V.; Likhoshvai, Vitaly A.; Fadeev, Stanislav I.; Khlebodarova, Tamara M.

We have investigated the scenarios of transition to chaos in the mathematical model of a genetic system constituted by a single transcription factor-encoding gene, the expression of which is self-regulated by a feedback loop that involves protein isoforms. Alternative splicing results in the synthesis of protein isoforms providing opposite regulatory outcomes — activation or repression. The model is represented by a differential equation with two delayed arguments. The possibility of transition to chaos dynamics via all classical scenarios: a cascade of period-doubling bifurcations, quasiperiodicity and type-I, type-II and type-III intermittencies, has been numerically demonstrated. The parametric features of each type of transition to chaos have been described.

A comparison between the stability properties in a DDE model for leukemia and the modified fractional counterpart

NASA Astrophysics Data System (ADS)

Rǎdulescu, I. R.; Cândea, D.; Kaslik, E.

2017-01-01

In this paper, a delay differential equations (DDEs) model of leukemia is introduced and its dynamical properties are investigated in comparison with the modified fractional-order system where the Caputo's derivative is used. The model takes into account three types of division that a stem-like cell can undergo and cell competition between healthy and leukemia cell populations. The action of the immune system on the leukemic cell populations is also considered. The stability properties of the equilibrium points are established through numerical results and the differences between the two types of approaches are discussed. Medical conclusions are drawn in view of the obtained numerical simulations.

Mathematical and Computational Modeling for Tumor Virotherapy with Mediated Immunity.

PubMed

Timalsina, Asim; Tian, Jianjun Paul; Wang, Jin

2017-08-01

We propose a new mathematical modeling framework based on partial differential equations to study tumor virotherapy with mediated immunity. The model incorporates both innate and adaptive immune responses and represents the complex interaction among tumor cells, oncolytic viruses, and immune systems on a domain with a moving boundary. Using carefully designed computational methods, we conduct extensive numerical simulation to the model. The results allow us to examine tumor development under a wide range of settings and provide insight into several important aspects of the virotherapy, including the dependence of the efficacy on a few key parameters and the delay in the adaptive immunity. Our findings also suggest possible ways to improve the virotherapy for tumor treatment.

Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory

NASA Astrophysics Data System (ADS)

Zhou, L.-Q.; Meleshko, S. V.

2017-07-01

The group analysis method is applied to a system of integro-differential equations corresponding to a linear thermoviscoelastic model. A recently developed approach for calculating the symmetry groups of such equations is used. The general solution of the determining equations for the system is obtained. Using subalgebras of the admitted Lie algebra, two classes of partially invariant solutions of the considered system of integro-differential equations are studied.

On the hierarchy of partially invariant submodels of differential equations

NASA Astrophysics Data System (ADS)

Golovin, Sergey V.

2008-07-01

It is noted that the partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PISs of the higher rank. This introduces a hierarchic structure in the set of all PISs of a given system of differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. The hierarchy simplifies the process of enumeration and analysis of partially invariant submodels to the given system of differential equations. In this framework, the complete classification of regular partially invariant solutions of ideal MHD equations is given.

Optimal moving grids for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Wathen, A. J.

1989-01-01

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of partial differential equation solutions in the least squares norm.

Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations

NASA Technical Reports Server (NTRS)

Geddes, K. O.

1977-01-01

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order

NASA Astrophysics Data System (ADS)

Tu, Jin; Yi, Cai-Feng

2008-04-01

In this paper, the authors investigate the growth of solutions of a class of higher order linear differential equationsf(k)+Ak-1f(k-1)+...+A0f=0 when most coefficients in the above equations have the same order with each other, and obtain some results which improve previous results due to K.H. Kwon [K.H. Kwon, Nonexistence of finite order solutions of certain second order linear differential equations, Kodai Math. J. 19 (1996) 378-387] and ZE-X. Chen [Z.-X. Chen, The growth of solutions of the differential equation f''+e-zf'+Q(z)f=0, Sci. China Ser. A 31 (2001) 775-784 (in Chinese); ZE-X. Chen, On the hyper order of solutions of higher order differential equations, Chinese Ann. Math. Ser. B 24 (2003) 501-508 (in Chinese); Z.-X. Chen, On the growth of solutions of a class of higher order differential equations, Acta Math. Sci. Ser. B 24 (2004) 52-60 (in Chinese); Z.-X. Chen, C.-C. Yang, Quantitative estimations on the zeros and growth of entire solutions of linear differential equations, Complex Var. 42 (2000) 119-133].

A delay differential model of ENSO variability: Extreme values and stability analysis

NASA Astrophysics Data System (ADS)

Zaliapin, I.; Ghil, M.

2009-04-01

We consider a delay differential equation (DDE) model for El-Niño Southern Oscillation (ENSO) variability [Ghil et al. (2008), Nonlin. Proc. Geophys., 15, 417-433.] The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. Toy models of this type were shown to capture major features of the ENSO phenomenon [Jin et al., Science (1994); Tziperman et al., Science (1994)]; they provide a convenient paradigm for explaining interannual ENSO variability and shed new light on its dynamical properties. So far, though, DDE model studies of ENSO have been limited to linear stability analysis of steady-state solutions, which are not typical in forced systems, case studies of particular trajectories, or one-dimensional scenarios of transition to chaos, varying a single parameter while the others are kept fixed. In this work we take several steps toward a comprehensive analysis of DDE models relevant for ENSO phenomenology and illustrate the complexity of phase-parameter space structure for even such a simple model of climate dynamics. We formulate an initial value problem for our model and prove the existence, uniqueness, and continuous dependence theorem. We then use this theoretical result to perform detailed numerical stability analyses of the model in the three-dimensional space of its physically relevant parameters: strength of seasonal forcing b, atmosphere-ocean coupling ΰ, and propagation period ? of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are reported; they are separated by a sharp neutral curve in the (b,?) plane at constant ΰ. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling ΰ increases. In the unstable regime, spontaneous transitions occur in the mean temperature (i.e., thermocline depth), period, and extreme annual values, for purely periodic, seasonal forcing. The model reproduces the Devils bleachers characterizing other ENSO models, such as nonlinear, coupled systems of partial differential equations; some of the features of this behavior have been documented in general circulation models, as well as in observations. We analyze the values of annual extremes and their location within an annual cycle and report the phase-locking phenomenon, which is connected to the occurrence of El-Niño events during the boreal (Northern Hemisphere) winter. We report existence of multiple solutions and study their basins of attraction in a space of initial conditions. We also present a model-based justification for the observed quasi-biennial oscillation in Tropical Pacific SSTs. We expect similar behavior in much more detailed and realistic models, where it is harder to describe its causes as completely. The basic mechanisms used in our model (delayed feedback and forcing) may be relevant to other natural systems in which internal instabilities interact with external forcing and give rise to extreme events.

Computation techniques and computer programs to analyze Stirling cycle engines using characteristic dynamic energy equations

NASA Technical Reports Server (NTRS)

Larson, V. H.

1982-01-01

The basic equations that are used to describe the physical phenomena in a Stirling cycle engine are the general energy equations and equations for the conservation of mass and conversion of momentum. These equations, together with the equation of state, an analytical expression for the gas velocity, and an equation for mesh temperature are used in this computer study of Stirling cycle characteristics. The partial differential equations describing the physical phenomena that occurs in a Stirling cycle engine are of the hyperbolic type. The hyperbolic equations have real characteristic lines. By utilizing appropriate points along these curved lines the partial differential equations can be reduced to ordinary differential equations. These equations are solved numerically using a fourth-fifth order Runge-Kutta integration technique.

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

NASA Astrophysics Data System (ADS)

Chelnokov, Yu. N.

2017-11-01

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo-Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper. A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues-Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given. Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo-Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass. The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.

Symbolic Solution of Linear Differential Equations

NASA Technical Reports Server (NTRS)

Feinberg, R. B.; Grooms, R. G.

1981-01-01

An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.

On differential operators generating iterative systems of linear ODEs of maximal symmetry algebra

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2017-06-01

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Periodicity and positivity of a class of fractional differential equations.

PubMed

Ibrahim, Rabha W; Ahmad, M Z; Mohammed, M Jasim

2016-01-01

Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.

Stochastic Evolution Equations Driven by Fractional Noises

DTIC Science & Technology

2016-11-28

rate of convergence to zero or the error and the limit in distribution of the error fluctuations. We have studied time discrete numerical schemes...error fluctuations. We have studied time discrete numerical schemes based on Taylor expansions for rough differential equations and for stochastic...variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian

Chandrasekhar equations for infinite dimensional systems

NASA Technical Reports Server (NTRS)

Ito, K.; Powers, R.

1985-01-01

The existence of Chandrasekhar equations for linear time-invariant systems defined on Hilbert spaces is investigated. An important consequence is that the solution to the evolutional Riccati equation is strongly differentiable in time, and that a strong solution of the Riccati differential equation can be defined. A discussion of the linear-quadratic optimal-control problem for hereditary differential systems is also included.

Traveling waves in a coupled reaction-diffusion and difference model of hematopoiesis

NASA Astrophysics Data System (ADS)

Adimy, M.; Chekroun, A.; Kazmierczak, B.

2017-04-01

The formation and development of blood cells is a very complex process, called hematopoiesis. This process involves a small population of cells called hematopoietic stem cells (HSCs). The HSCs are undifferentiated cells, located in the bone marrow before they become mature blood cells and enter the blood stream. They have a unique ability to produce either similar cells (self-renewal), or cells engaged in one of different lineages of blood cells: red blood cells, white cells and platelets (differentiation). The HSCs can be either in a proliferating or in a quiescent phase. In this paper, we distinguish between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. We propose a mathematical model describing the dynamics of HSC population, taking into account their spatial distribution. The resulting model is a coupled reaction-diffusion equation and difference equation with delay. We study the existence of monotone traveling wave fronts and the asymptotic speed of spread.

Outcomes of a service teaching module on ODEs for physics students

NASA Astrophysics Data System (ADS)

Hyland, Diarmaid; van Kampen, Paul; Nolan, Brien C.

2018-07-01

This paper reports on the first part of a multiphase research project that seeks to identify and address the difficulties encountered by physics students when studying differential equations. Differential equations are used extensively by undergraduate physics students, particularly in the advanced modules of their degree. It is, therefore, necessary that students develop conceptual understanding of differential equations in addition to procedural skills. We have investigated the difficulties encountered by third-year students at Dublin City University in an introductory differential equations module. We developed a survey to identify these difficulties and administered it to students who had recently completed the module. We found that students' mathematical ability in relation to procedural competence is an issue in their study of differential equations, but not as severe an issue as their conceptual understanding. Mathematical competence alone is insufficient if we expect our students to be able to recognize the need for differential equations in a physical context and to be able to set up, solve and interpret the solutions of such equations. We discuss the implications of these results for the next stages of the research project.

Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions

NASA Astrophysics Data System (ADS)

Ding, Xiao-Li; Nieto, Juan J.

2017-11-01

In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

The method of Ritz applied to the equation of Hamilton. [for pendulum systems

NASA Technical Reports Server (NTRS)

Bailey, C. D.

1976-01-01

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations

NASA Astrophysics Data System (ADS)

Demina, Maria V.; Kudryashov, Nikolay A.

2011-03-01

Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational, periodic, elliptic) are found for a wide class of autonomous nonlinear ordinary differential equations.

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

DOE PAGES

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris; ...

2018-04-23

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching.

PubMed

Chow, Sy-Miin; Ou, Lu; Ciptadi, Arridhana; Prince, Emily B; You, Dongjun; Hunter, Michael D; Rehg, James M; Rozga, Agata; Messinger, Daniel S

2018-06-01

A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of [Formula: see text] mother-infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children's tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.

On the systematic approach to the classification of differential equations by group theoretical methods

NASA Astrophysics Data System (ADS)

Andriopoulos, K.; Dimas, S.; Leach, P. G. L.; Tsoubelis, D.

2009-08-01

Complete symmetry groups enable one to characterise fully a given differential equation. By considering the reversal of an approach based upon complete symmetry groups we construct new classes of differential equations which have the equations of Bateman, Monge-Ampère and Born-Infeld as special cases. We develop a symbolic algorithm to decrease the complexity of the calculations involved.

Solving Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1987-01-01

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Chandrasekhar equations for infinite dimensional systems

NASA Technical Reports Server (NTRS)

Ito, K.; Powers, R. K.

1985-01-01

Chandrasekhar equations are derived for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati equation is strongly differentiable in time and one can define a strong solution of the Riccati differential equation. A detailed discussion on the linear quadratic optimal control problem for hereditary differential systems is also included.

A representation of solution of stochastic differential equations

NASA Astrophysics Data System (ADS)

Kim, Yoon Tae; Jeon, Jong Woo

2006-03-01

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

NASA Astrophysics Data System (ADS)

Assanova, A. T.; Imanchiyev, A. E.; Kadirbayeva, Zh. M.

2018-04-01

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge-Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.

Generalized Lie symmetry approach for fractional order systems of differential equations. III

NASA Astrophysics Data System (ADS)

Singla, Komal; Gupta, R. K.

2017-06-01

The generalized Lie symmetry technique is proposed for the derivation of point symmetries for systems of fractional differential equations with an arbitrary number of independent as well as dependent variables. The efficiency of the method is illustrated by its application to three higher dimensional nonlinear systems of fractional order partial differential equations consisting of the (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov system, (3 + 1)-dimensional Burgers system, and (3 + 1)-dimensional Navier-Stokes equations. With the help of derived Lie point symmetries, the corresponding invariant solutions transform each of the considered systems into a system of lower-dimensional fractional partial differential equations.

Structure of Lie point and variational symmetry algebras for a class of odes

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2018-04-01

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

NASA Astrophysics Data System (ADS)

Filimonov, M. Yu.

2017-12-01

The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed.

Transformation matrices between non-linear and linear differential equations

NASA Technical Reports Server (NTRS)

Sartain, R. L.

1983-01-01

In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.

Programmable Differential Delay Circuit With Fine Delay Adjustment

DOEpatents

DeRyckere, John F.; Jenkins, Philip Nord; Cornett, Frank Nolan

2002-07-09

Circuitry that provides additional delay to early arriving signals such that all data signals arrive at a receiving latch with same path delay. The delay of a forwarded clock reference is also controlled such that the capturing clock edge will be optimally positioned near quadrature (depending on latch setup/hold requirements). The circuitry continuously adapts to data and clock path delay changes and digital filtering of phase measurements reduce errors brought on by jittering data edges. The circuitry utilizes only the minimum amount of delay necessary to achieve objective thereby limiting any unintended jitter. Particularly, this programmable differential delay circuit with fine delay adjustment is designed to allow the skew between ASICS to be minimized. This includes skew between data bits, between data bits and clocks as well as minimizing the overall skew in a channel between ASICS.

Some Theoretical Aspects of Nonzero Sum Differential Games and Applications to Combat Problems

DTIC Science & Technology

1971-06-01

the Equilibrium Solution . 7 Hamilton-Jacobi-Bellman Partial Differential Equations ............. .............. 9 Influence Function Differential...Linearly .......... ............ 18 Problem Statement .......... ............ 18 Formulation of LJB Equations, Influence Function Equations and the TPBVP...19 Control Lawe . . .. ...... ........... 21 Conditions for Influence Function Continuity along Singular Surfaces

Lie algebras and linear differential equations.

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

Solving Differential Equations Using Modified Picard Iteration

ERIC Educational Resources Information Center

Robin, W. A.

2010-01-01

Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

Ordinary differential equation for local accumulation time.

PubMed

Berezhkovskii, Alexander M

2011-08-21

Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential equation. Using this equation one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential equation, as was done in previous studies. We derive this ordinary differential equation together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this equation. © 2011 American Institute of Physics

Nonlinear waves of a nonlocal modified KdV equation in the atmospheric and oceanic dynamical system

NASA Astrophysics Data System (ADS)

Tang, Xiao-yan; Liang, Zu-feng; Hao, Xia-zhi

2018-07-01

A new general nonlocal modified KdV equation is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a β-plane. The nonlocal property is manifested in the shifted parity and delayed time reversal symmetries. Exact solutions of the nonlocal modified KdV equation are obtained including periodic waves, kink waves, solitary waves, kink- and/or anti-kink-cnoidal periodic wave interaction solutions, which can be utilized to describe various two-place and time-delayed correlated events. As an illustration, a special approximate solution is applied to theoretically capture the salient features of two correlated dipole blocking events in atmospheric dynamical systems.

Transforming parts of a differential equations system to difference equations as a method for run-time savings in NONMEM.

PubMed

Petersson, K J F; Friberg, L E; Karlsson, M O

2010-10-01

Computer models of biological systems grow more complex as computing power increase. Often these models are defined as differential equations and no analytical solutions exist. Numerical integration is used to approximate the solution; this can be computationally intensive, time consuming and be a large proportion of the total computer runtime. The performance of different integration methods depend on the mathematical properties of the differential equations system at hand. In this paper we investigate the possibility of runtime gains by calculating parts of or the whole differential equations system at given time intervals, outside of the differential equations solver. This approach was tested on nine models defined as differential equations with the goal to reduce runtime while maintaining model fit, based on the objective function value. The software used was NONMEM. In four models the computational runtime was successfully reduced (by 59-96%). The differences in parameter estimates, compared to using only the differential equations solver were less than 12% for all fixed effects parameters. For the variance parameters, estimates were within 10% for the majority of the parameters. Population and individual predictions were similar and the differences in OFV were between 1 and -14 units. When computational runtime seriously affects the usefulness of a model we suggest evaluating this approach for repetitive elements of model building and evaluation such as covariate inclusions or bootstraps.

Single generation cycles and delayed feedback cycles are not separate phenomena.

PubMed

Pfaff, T; Brechtel, A; Drossel, B; Guill, C

2014-12-01

We study a simple model for generation cycles, which are oscillations with a period of one or a few generation times of the species. The model is formulated in terms of a single delay-differential equation for the population density of an adult stage, with recruitment to the adult stage depending on the intensity of competition during the juvenile phase. This model is a simplified version of a group of models proposed by Gurney and Nisbet, who were the first to distinguish between single-generation cycles and delayed-feedback cycles. According to these authors, the two oscillation types are caused by different mechanisms and have periods in different intervals, which are one to two generation times for single-generation cycles and two to four generation times for delayed-feedback cycles. By abolishing the strict coupling between the maturation time and the time delay between competition and its effect on the population dynamics, we find that single-generation cycles and delayed-feedback cycles occur in the same model version, with a gradual transition between the two as the model parameters are varied over a sufficiently large range. Furthermore, cycle periods are not bounded to lie within single octaves. This implies that a clear distinction between different types of generation cycles is not possible. Cycles of all periods and even chaos can be generated by varying the parameters that determine the time during which individuals from different cohorts compete with each other. This suggests that life-cycle features in the juvenile stage and during the transition to the adult stage are important determinants of the dynamics of density limited populations. Copyright © 2014 Elsevier Inc. All rights reserved.

General existence principles for Stieltjes differential equations with applications to mathematical biology

NASA Astrophysics Data System (ADS)

López Pouso, Rodrigo; Márquez Albés, Ignacio

2018-04-01

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.

Symmetry investigations on the incompressible stationary axisymmetric Euler equations with swirl

NASA Astrophysics Data System (ADS)

Frewer, M.; Oberlack, M.; Guenther, S.

2007-08-01

We discuss the incompressible stationary axisymmetric Euler equations with swirl, for which we derive via a scalar stream function an equivalent representation, the Bragg-Hawthorne equation [Bragg, S.L., Hawthorne, W.R., 1950. Some exact solutions of the flow through annular cascade actuator discs. J. Aero. Sci. 17, 243]. Despite this obvious equivalence, we will show that under a local Lie point symmetry analysis the Bragg-Hawthorne equation exposes itself as not being fully equivalent to the original Euler equations. This is reflected in the way that it possesses additional symmetries not being admitted by its counterpart. In other words, a symmetry of the Bragg-Hawthorne equation is in general not a symmetry of the Euler equations. Not the differential Euler equations but rather a set of integro-differential equations attains full equivalence to the Bragg-Hawthorne equation. For these intermediate Euler equations, it is interesting to note that local symmetries of the Bragg-Hawthorne equation transform to local as well as to nonlocal symmetries. This behaviour, on the one hand, is in accordance with Zawistowski's result [Zawistowski, Z.J., 2001. Symmetries of integro-differential equations. Rep. Math. Phys. 48, 269; Zawistowski, Z.J., 2004. General criterion of invariance for integro-differential equations. Rep. Math. Phys. 54, 341] that it is possible for integro-differential equations to admit local Lie point symmetries. On the other hand, with this transformation process we collect symmetries which cannot be obtained when carrying out a usual local Lie point symmetry analysis. Finally, the symmetry classification of the Bragg-Hawthorne equation is used to find analytical solutions for the phenomenon of vortex breakdown.

Speed Measurement and Motion Analysis of Chang'E-3 Rover Based on Differential Phase Delay

NASA Astrophysics Data System (ADS)

Pan, C.; Liu, Q. H.; Zheng, X.; He, Q. B.; Wu, Y. J.

2015-07-01

On 2013 December 14, the Chang'E-3 made a successful soft landing on the lunar surface, and then carried out the tasks of separating the lander and the rover, and taking the photos of each other. With the same beam VLBI (Very long baseline interferometry) technique to observe the signals transmitted by the lander and the rover simultaneously, the differential phase delay between them is calculated, which can reflect a minor change of the rover's position on a scale of a few centimeters. Based on the high sensitivity of differential phase delay, the rover's speeds during 5 movements are obtained with an average of 0.056 m/s. The relationship between the rover's shake in moving process, and lunar terrain is analyzed by using the spectrum of the residual of the differential phase delay after the first-order polynomial fitting.

Speed Measurement and Motion Analysis of Chang'E-3 Rover Based on Differential Phase Delay

NASA Astrophysics Data System (ADS)

Chao, Pan; Qing-hui, Liu; Xin, Zheng; Qing-bao, He; Ya-jun, Wu

2016-04-01

On 14th December 2013, the Chang'E-3 made a successful soft landing on the lunar surface, and then carried out the tasks of separating the lander and the rover, and taking pictures of each other. With the same beam VLBI (Very Long Baseline Interferometry) technique to observe the signals transmitted by the lander and the rover simultaneously, the differential phase delay between them is calculated, which can reflect the minor changes of the rover's position on a scale of a few centimeters. Based on the high sensitivity of differential phase delay, the rover's speeds during 5 movements are obtained with an average of 0.056 m/s. The relationship between the rover's shake in the moving process and the lunar terrain is analyzed by using the spectrum of the residual of the differential phase delay after the first-order polynomial fitting.

A note on the generation of phase plane plots on a digital computer. [for solution of nonlinear differential equations

NASA Technical Reports Server (NTRS)

Simon, M. K.

1980-01-01

A technique is presented for generating phase plane plots on a digital computer which circumvents the difficulties associated with more traditional methods of numerical solving nonlinear differential equations. In particular, the nonlinear differential equation of operation is formulated.

MACSYMA's symbolic ordinary differential equation solver

NASA Technical Reports Server (NTRS)

Golden, J. P.

1977-01-01

The MACSYMA's symbolic ordinary differential equation solver ODE2 is described. The code for this routine is delineated, which is of interest because it is written in top-level MACSYMA language, and may serve as a good example of programming in that language. Other symbolic ordinary differential equation solvers are mentioned.

Undergraduate Students' Mental Operations in Systems of Differential Equations

ERIC Educational Resources Information Center

Whitehead, Karen; Rasmussen, Chris

2003-01-01

This paper reports on research conducted to understand undergraduate students' ways of reasoning about systems of differential equations (SDEs). As part of a semester long classroom teaching experiment in a first course in differential equations, we conducted task-based interviews with six students after their study of first order differential…

Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

ERIC Educational Resources Information Center

Deboeck, Pascal R.; Boker, Steven M.

2010-01-01

Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for…

Variable-mesh method of solving differential equations

NASA Technical Reports Server (NTRS)

Van Wyk, R.

1969-01-01

Multistep predictor-corrector method for numerical solution of ordinary differential equations retains high local accuracy and convergence properties. In addition, the method was developed in a form conducive to the generation of effective criteria for the selection of subsequent step sizes in step-by-step solution of differential equations.

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

NASA Technical Reports Server (NTRS)

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)

2002-01-01

We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.

Given a one-step numerical scheme, on which ordinary differential equations is it exact?

NASA Astrophysics Data System (ADS)

Villatoro, Francisco R.

2009-01-01

A necessary condition for a (non-autonomous) ordinary differential equation to be exactly solved by a one-step, finite difference method is that the principal term of its local truncation error be null. A procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed. Examples of differential equations exactly solved by the explicit Euler, implicit Euler, trapezoidal rule, second-order Taylor, third-order Taylor, van Niekerk's second-order rational, and van Niekerk's third-order rational methods are presented.

Ordinary differential equations with applications in molecular biology.

PubMed

Ilea, M; Turnea, M; Rotariu, M

2012-01-01

Differential equations are of basic importance in molecular biology mathematics because many biological laws and relations appear mathematically in the form of a differential equation. In this article we presented some applications of mathematical models represented by ordinary differential equations in molecular biology. The vast majority of quantitative models in cell and molecular biology are formulated in terms of ordinary differential equations for the time evolution of concentrations of molecular species. Assuming that the diffusion in the cell is high enough to make the spatial distribution of molecules homogenous, these equations describe systems with many participating molecules of each kind. We propose an original mathematical model with small parameter for biological phospholipid pathway. All the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. If we reduce the size of the solution region the same small epsilon will result in a different condition number. It is clear that the solution for a smaller region is less difficult. We introduce the mathematical technique known as boundary function method for singular perturbation system. In this system, the small parameter is an asymptotic variable, different from the independent variable. In general, the solutions of such equations exhibit multiscale phenomena. Singularly perturbed problems form a special class of problems containing a small parameter which may tend to zero. Many molecular biology processes can be quantitatively characterized by ordinary differential equations. Mathematical cell biology is a very active and fast growing interdisciplinary area in which mathematical concepts, techniques, and models are applied to a variety of problems in developmental medicine and bioengineering. Among the different modeling approaches, ordinary differential equations (ODE) are particularly important and have led to significant advances. Ordinary differential equations are used to model biological processes on various levels ranging from DNA molecules or biosynthesis phospholipids on the cellular level.

Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials

NASA Astrophysics Data System (ADS)

Volkmer, Hans

2008-04-01

Sequences of polynomials, orthogonal with respect to signed measures, are associated with a class of differential equations including the Mathieu, Lame and Whittaker-Hill equation. It is shown that the zeros of pn form sequences which converge to the eigenvalues of the corresponding differential equations. Moreover, interlacing properties of the zeros of pn are found. Applications to the numerical treatment of eigenvalue problems are given.

On the Solution of Elliptic Partial Differential Equations on Regions with Corners

DTIC Science & Technology

2015-07-09

In this report we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations . We observe...that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of...efficient numerical algorithms. The results are illustrated by a number of numerical examples. On the solution of elliptic partial differential equations on

State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities.

PubMed

Korayem, M H; Nekoo, S R

2015-07-01

This work studies an optimal control problem using the state-dependent Riccati equation (SDRE) in differential form to track for time-varying systems with state and control nonlinearities. The trajectory tracking structure provides two nonlinear differential equations: the state-dependent differential Riccati equation (SDDRE) and the feed-forward differential equation. The independence of the governing equations and stability of the controller are proven along the trajectory using the Lyapunov approach. Backward integration (BI) is capable of solving the equations as a numerical solution; however, the forward solution methods require the closed-form solution to fulfill the task. A closed-form solution is introduced for SDDRE, but the feed-forward differential equation has not yet been obtained. Different ways of solving the problem are expressed and analyzed. These include BI, closed-form solution with corrective assumption, approximate solution, and forward integration. Application of the tracking problem is investigated to control robotic manipulators possessing rigid or flexible joints. The intention is to release a general program for automatic implementation of an SDDRE controller for any manipulator that obeys the Denavit-Hartenberg (D-H) principle when only D-H parameters are received as input data. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.

The numerical solution of linear multi-term fractional differential equations: systems of equations

NASA Astrophysics Data System (ADS)

Edwards, John T.; Ford, Neville J.; Simpson, A. Charles

2002-11-01

In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.

Sparse dynamics for partial differential equations

PubMed Central

Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D.; Osher, Stanley

2013-01-01

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms. PMID:23533273

Exact solutions for STO and (3+1)-dimensional KdV-ZK equations using (G‧/G2) -expansion method

NASA Astrophysics Data System (ADS)

Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Ullah, Rahmat; Ahmed, Naveed; Khan, Umar

This article deals with finding some exact solutions of nonlinear fractional differential equations (NLFDEs) by applying a relatively new method known as (G‧/G2) -expansion method. Solutions of space-time fractional Sharma-Tasso-Olever (STO) equation of fractional order and (3+1)-dimensional KdV-Zakharov Kuznetsov (KdV-ZK) equation of fractional order are reckoned to demonstrate the validity of this method. The fractional derivative version of modified Riemann-Liouville, linked with Fractional complex transform is employed to transform fractional differential equations into the corresponding ordinary differential equations.

Sparse dynamics for partial differential equations.

PubMed

Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D; Osher, Stanley

2013-04-23

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.

Lattice Boltzmann model for high-order nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

PubMed

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Mathematics for Physics

NASA Astrophysics Data System (ADS)

Stone, Michael; Goldbart, Paul

2009-07-01

Preface; 1. Calculus of variations; 2. Function spaces; 3. Linear ordinary differential equations; 4. Linear differential operators; 5. Green functions; 6. Partial differential equations; 7. The mathematics of real waves; 8. Special functions; 9. Integral equations; 10. Vectors and tensors; 11. Differential calculus on manifolds; 12. Integration on manifolds; 13. An introduction to differential topology; 14. Group and group representations; 15. Lie groups; 16. The geometry of fibre bundles; 17. Complex analysis I; 18. Applications of complex variables; 19. Special functions and complex variables; Appendixes; Reference; Index.

An Estimation Theory for Differential Equations and other Problems, with Applications.

DTIC Science & Technology

1981-11-01

order differential -8- operators and M-operators, in particular, the Perron - Frobenius theory and generalizations. Convergence theory for iterative... THEORY FOR DIFFERENTIAL 0EQUATIONS AND OTHER FROBLEMS, WITH APPLICATIONS 0 ,Final Technical Report by Johann Schr6der November, 1981 EUROPEAN RESEARCH...COVERED An estimation theory for differential equations Final Report and other problrms, with app)lications A981 6. PERFORMING ORG. RN,-ORT NUMfFR 7

Dynamics and Control of Constrained Multibody Systems modeled with Maggi's equation: Application to Differential Mobile Robots Part I

NASA Astrophysics Data System (ADS)

Amengonu, Yawo H.; Kakad, Yogendra P.

2014-07-01

Quasivelocity techniques such as Maggi's and Boltzmann-Hamel's equations eliminate Lagrange multipliers from the beginning as opposed to the Euler-Lagrange method where one has to solve for the n configuration variables and the multipliers as functions of time when there are m nonholonomic constraints. Maggi's equation produces n second-order differential equations of which (n-m) are derived using (n-m) independent quasivelocities and the time derivative of the m kinematic constraints which add the remaining m second order differential equations. This technique is applied to derive the dynamics of a differential mobile robot and a controller which takes into account these dynamics is developed.

The differential equation of an arbitrary reflecting surface

NASA Astrophysics Data System (ADS)

Melka, Richard F.; Berrettini, Vincent D.; Yousif, Hashim A.

2018-05-01

A differential equation describing the reflection of a light ray incident upon an arbitrary reflecting surface is obtained using the law of reflection. The derived equation is written in terms of a parameter and the value of this parameter determines the nature of the reflecting surface. Under various parametric constraints, the solution of the differential equation leads to the various conic surfaces but is not generally solvable. In addition, the dynamics of the light reflections from the conic surfaces are executed in the Mathematica software. Our derivation is the converse of the traditional approach and our analysis assumes a relation between the object distance and the image distance. This leads to the differential equation of the reflecting surface.

Two-dimensional integrating matrices on rectangular grids. [solving differential equations associated with rotating structures

NASA Technical Reports Server (NTRS)

Lakin, W. D.

1981-01-01

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

Intrinsic delay of permeable base transistor

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, Wenchao; Guo, Jing; So, Franky

2014-07-28

Permeable base transistors (PBTs) fabricated by vacuum deposition or solution process have the advantages of easy fabrication and low power operation and are a promising device structure for flexible electronics. Intrinsic delay of PBT, which characterizes the speed of the transistor, is investigated by solving the three-dimensional Poisson equation and drift-diffusion equation self-consistently using finite element method. Decreasing the emitter thickness lowers the intrinsic delay by improving on-current, and a thinner base is also preferred for low intrinsic delay because of fewer carriers in the base region at off-state. The intrinsic delay exponentially decreases as the emitter contact Schottky barriermore » height decreases, and it linearly depends on the carrier mobility. With an optimized emitter contact barrier height and device geometry, a sub-nano-second intrinsic delay can be achieved with a carrier mobility of ∼10 cm{sup 2}/V/s obtainable in solution processed indium gallium zinc oxide, which indicates the potential of solution processed PBTs for GHz operations.« less

On some control problems of dynamic of reactor

NASA Astrophysics Data System (ADS)

Baskakov, A. V.; Volkov, N. P.

2017-12-01

The paper analyzes controllability of the transient processes in some problems of nuclear reactor dynamics. In this case, the mathematical model of nuclear reactor dynamics is described by a system of integro-differential equations consisting of the non-stationary anisotropic multi-velocity kinetic equation of neutron transport and the balance equation of delayed neutrons. The paper defines the formulation of the linear problem on control of transient processes in nuclear reactors with application of spatially distributed actions on internal neutron sources, and the formulation of the nonlinear problems on control of transient processes with application of spatially distributed actions on the neutron absorption coefficient and the neutron scattering indicatrix. The required control actions depend on the spatial and velocity coordinates. The theorems on existence and uniqueness of these control actions are proved in the paper. To do this, the control problems mentioned above are reduced to equivalent systems of integral equations. Existence and uniqueness of the solution for this system of integral equations is proved by the method of successive approximations, which makes it possible to construct an iterative scheme for numerical analyses of transient processes in a given nuclear reactor with application of the developed mathematical model. Sufficient conditions for controllability of transient processes are also obtained. In conclusion, a connection is made between the control problems and the observation problems, which, by to the given information, allow us to reconstruct either the function of internal neutron sources, or the neutron absorption coefficient, or the neutron scattering indicatrix....

DOE Office of Scientific and Technical Information (OSTI.GOV)

Croft, Stephen; Favalli, Andrea

Here, we extend the familiar Bӧhnel point-model equations, which are routinely used to interpret neutron coincidence counting rates, by including the contribution of delayed neutrons. After developing the necessary equations we use them to show, by providing some numerical results, what the quantitative impact of neglecting delayed neutrons is across the full range of practical nuclear safeguards applications. The influence of delayed neutrons is predicted to be small for the types of deeply sub-critical assay problems which concern the nuclear safeguards community, smaller than uncertainties arising from other factors. This is most clearly demonstrated by considering the change in themore » effective (α,n)-to-spontaneous fission prompt-neutron ratio that the inclusion of delayed neutrons gives rise to. That the influence of delayed neutrons is small is fortunate, and our results justify the long standing practice of simply neglecting them in the analysis of field measurements.« less

Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system.

PubMed

Gurevich, Svetlana V

2014-10-28

The dynamics of a single breathing localized structure in a three-component reaction-diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov-Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner. © 2014 The Author(s) Published by the Royal Society. All rights reserved.

Back-propagation learning of infinite-dimensional dynamical systems.

PubMed

Tokuda, Isao; Tokunaga, Ryuji; Aihara, Kazuyuki

2003-10-01

This paper presents numerical studies of applying back-propagation learning to a delayed recurrent neural network (DRNN). The DRNN is a continuous-time recurrent neural network having time delayed feedbacks and the back-propagation learning is to teach spatio-temporal dynamics to the DRNN. Since the time-delays make the dynamics of the DRNN infinite-dimensional, the learning algorithm and the learning capability of the DRNN are different from those of the ordinary recurrent neural network (ORNN) having no time-delays. First, two types of learning algorithms are developed for a class of DRNNs. Then, using chaotic signals generated from the Mackey-Glass equation and the Rössler equations, learning capability of the DRNN is examined. Comparing the learning algorithms, learning capability, and robustness against noise of the DRNN with those of the ORNN and time delay neural network, advantages as well as disadvantages of the DRNN are investigated.

Sourcing for Parameter Estimation and Study of Logistic Differential Equation

ERIC Educational Resources Information Center

Winkel, Brian J.

2012-01-01

This article offers modelling opportunities in which the phenomena of the spread of disease, perception of changing mass, growth of technology, and dissemination of information can be described by one differential equation--the logistic differential equation. It presents two simulation activities for students to generate real data, as well as…

Differential equations for loop integrals in Baikov representation

NASA Astrophysics Data System (ADS)

Bosma, Jorrit; Larsen, Kasper J.; Zhang, Yang

2018-05-01

We present a proof that differential equations for Feynman loop integrals can always be derived in Baikov representation without involving dimension-shift identities. We moreover show that in a large class of two- and three-loop diagrams it is possible to avoid squared propagators in the intermediate steps of setting up the differential equations.

Remarks on the Non-Linear Differential Equation the Second Derivative of Theta Plus A Sine Theta Equals 0.

ERIC Educational Resources Information Center

Fay, Temple H.; O'Neal, Elizabeth A.

1985-01-01

The authors draw together a variety of facts concerning a nonlinear differential equation and compare the exact solution with approximate solutions. Then they provide an expository introduction to the elliptic sine function suitable for presentation in undergraduate courses on differential equations. (MNS)

Operator Factorization and the Solution of Second-Order Linear Ordinary Differential Equations

ERIC Educational Resources Information Center

Robin, W.

2007-01-01

The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting…

Monograph - The Numerical Integration of Ordinary Differential Equations.

ERIC Educational Resources Information Center

Hull, T. E.

The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…

The Local Brewery: A Project for Use in Differential Equations Courses

ERIC Educational Resources Information Center

Starling, James K.; Povich, Timothy J.; Findlay, Michael

2016-01-01

We describe a modeling project designed for an ordinary differential equations (ODEs) course using first-order and systems of first-order differential equations to model the fermentation process in beer. The project aims to expose the students to the modeling process by creating and solving a mathematical model and effectively communicating their…

An Engineering-Oriented Approach to the Introductory Differential Equations Course

ERIC Educational Resources Information Center

Pennell, S.; Avitabile, P.; White, J.

2009-01-01

The introductory differential equations course can be made more relevant to engineering students by including more of the engineering viewpoint, in which differential equations are regarded as systems with inputs and outputs. This can be done without sacrificing any of the usual topical coverage. This point of view is conducive to student…

Dynamics of the Pin Pallet Runaway Escapement

DTIC Science & Technology

1978-06-01

for Continued Work 29 References 32 I Appendixes A Kinematics of Coupled Motion 34 B Differential Equation of Coupled Motion 38 f C Moment Arms 42 D...Expressions for these quantities are derived in appendix D. The differential equations for the free motion of the pallet and the escape-wheel are...Coupled Motion (location 100) To solve the differential equation of coupled motion (see equation .B (-10) of appendix B)- the main program calls on

Real-time optical laboratory solution of parabolic differential equations

NASA Technical Reports Server (NTRS)

Casasent, David; Jackson, James

1988-01-01

An optical laboratory matrix-vector processor is used to solve parabolic differential equations (the transient diffusion equation with two space variables and time) by an explicit algorithm. This includes optical matrix-vector nonbase-2 encoded laboratory data, the combination of nonbase-2 and frequency-multiplexed data on such processors, a high-accuracy optical laboratory solution of a partial differential equation, new data partitioning techniques, and a discussion of a multiprocessor optical matrix-vector architecture.

Stochastic differential equation model for linear growth birth and death processes with immigration and emigration

DOE Office of Scientific and Technical Information (OSTI.GOV)

Granita, E-mail: granitafc@gmail.com; Bahar, A.

This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.

Liouvillian propagators, Riccati equation and differential Galois theory

NASA Astrophysics Data System (ADS)

Acosta-Humánez, Primitivo; Suazo, Erwin

2013-11-01

In this paper a Galoisian approach to building propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schrödinger equation and the virtual solvability of the differential Galois group of its associated characteristic equation. As the main application of this approach we solve Ince’s differential equation through the Hamiltonian algebrization procedure and the Kovacic algorithm to find the propagator for a generalized harmonic oscillator. This propagator has applications which describe the process of degenerate parametric amplification in quantum optics and light propagation in a nonlinear anisotropic waveguide. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.

The condition of regular degeneration for singularly perturbed systems of linear differential-difference equations.

NASA Technical Reports Server (NTRS)

Cooke, K. L.; Meyer, K. R.

1966-01-01

Extension of problem of singular perturbation for linear scalar constant coefficient differential- difference equation with single retardation to several retardations, noting degenerate equation solution

Continuous angle steering of an optically- controlled phased array antenna based on differential true time delay constituted by micro-optical components.

PubMed

Wang, Jian; Hou, Peipei; Cai, Haiwen; Sun, Jianfeng; Wang, Shunan; Wang, Lijuan; Yang, Fei

2015-04-06

We propose an optically controlled phased array antenna (PAA) based on differential true time delay constructed optical beamforming network (OBFN). Differential true time delay is realized by stack integrated micro-optical components. Optically-controlled angle steering of radio frequency (RF) beams are realized and demonstrated by this configuration. Experimental results demonstrate that OBFN based PAA can accomplish RF-independent broadband beam steering without beam squint effect and can achieve continuous angle steering. In addition, multi-beams for different steering angles are acquired synchronously.

Estimating Soil Hydraulic Parameters using Gradient Based Approach

NASA Astrophysics Data System (ADS)

Rai, P. K.; Tripathi, S.

2017-12-01

The conventional way of estimating parameters of a differential equation is to minimize the error between the observations and their estimates. The estimates are produced from forward solution (numerical or analytical) of differential equation assuming a set of parameters. Parameter estimation using the conventional approach requires high computational cost, setting-up of initial and boundary conditions, and formation of difference equations in case the forward solution is obtained numerically. Gaussian process based approaches like Gaussian Process Ordinary Differential Equation (GPODE) and Adaptive Gradient Matching (AGM) have been developed to estimate the parameters of Ordinary Differential Equations without explicitly solving them. Claims have been made that these approaches can straightforwardly be extended to Partial Differential Equations; however, it has been never demonstrated. This study extends AGM approach to PDEs and applies it for estimating parameters of Richards equation. Unlike the conventional approach, the AGM approach does not require setting-up of initial and boundary conditions explicitly, which is often difficult in real world application of Richards equation. The developed methodology was applied to synthetic soil moisture data. It was seen that the proposed methodology can estimate the soil hydraulic parameters correctly and can be a potential alternative to the conventional method.

Differential Equations Models to Study Quorum Sensing.

PubMed

Pérez-Velázquez, Judith; Hense, Burkhard A

2018-01-01

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Delay in Apoptosome Formation Attenuates Apoptosis in Mouse Embryonic Stem Cell Differentiation

PubMed Central

Akbari-Birgani, Shiva; Hosseinkhani, Saman; Mollamohamadi, Sepideh; Baharvand, Hossein

2014-01-01

Differentiation is an inseparable process of development in multicellular organisms. Mouse embryonic stem cells (mESCs) represent a valuable research tool to conduct in vitro studies of cell differentiation. Apoptosis as a well known cell death mechanism shows some common features with cell differentiation, which has caused a number of ambiguities in the field. The research question here is how cells could differentiate these two processes from each other. We have investigated the role of the mitochondrial apoptotic pathway and cell energy level during differentiation of mESCs into the cardiomyocytes and their apoptosis. p53 expression, cytochrome c release, apoptosome formation, and caspase-3/7 activation are observed upon induction of both apoptosis and differentiation. However, remarkable differences are detected in time of cytochrome c appearance, apoptosome formation, and caspase activity upon induction of both processes. In apoptosis, apoptosome formation and caspase activity were observed rapidly following the cytochrome c release. Unlike apoptosis, the release of cytochrome c upon differentiation took more time, and the maximum caspase activity was also postponed for 24 h. This delay suggests that there is a regulatory mechanism during differentiation of mESCs into cardiomyocytes. The highest ATP content of cells was observed immediately after cytochrome c release 6 h after apoptosis induction and then decreased, but it was gradually increased up to 48 h after differentiation. These observations suggest that a delay in the release of cytochrome c or delay in ATP increase attenuate apoptosome formation, and caspase activation thereby discriminates apoptosis from differentiation in mESCs. PMID:24755221

On the singular perturbations for fractional differential equation.

PubMed

Atangana, Abdon

2014-01-01

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

Linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2011-06-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and the variation of constants method. The approach presented here can be used in a first course on differential equations for science and engineering majors.

Scabies in residential care homes: Modelling, inference and interventions for well-connected population sub-units

PubMed Central

Middleton, Jo; Güttel, Stefan; Cassell, Jackie; Ross, Joshua

2018-01-01

In the context of an ageing population, understanding the transmission of infectious diseases such as scabies through well-connected sub-units of the population, such as residential care homes, is particularly important for the design of efficient interventions to mitigate against the effects of those diseases. Here, we present a modelling methodology based on the efficient solution of a large-scale system of linear differential equations that allows statistical calibration of individual-based random models to real data on scabies in residential care homes. In particular, we review and benchmark different numerical methods for the integration of the differential equation system, and then select the most appropriate of these methods to perform inference using Markov chain Monte Carlo. We test the goodness-of-fit of this model using posterior predictive intervals and propagate forward the resulting parameter uncertainty in a Bayesian framework to consider the economic cost of delayed interventions against scabies, quantifying the benefits of prompt action in the event of detection. We also revisit the previous methodology used to assess the safety of treatments in small population sub-units—in this context ivermectin—and demonstrate that even a very slight relaxation of the implicit assumption of homogeneous death rates significantly increases the plausibility of the hypothesis that ivermectin does not cause excess mortality based upon the data of Barkwell and Shields. PMID:29579037

Newton's method: A link between continuous and discrete solutions of nonlinear problems

NASA Technical Reports Server (NTRS)

Thurston, G. A.

1980-01-01

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Choi, Cheong R.

The structural changes of kinetic Alfvén solitary waves (KASWs) due to higher-order terms are investigated. While the first-order differential equation for KASWs provides the dispersion relation for kinetic Alfvén waves, the second-order differential equation describes the structural changes of the solitary waves due to higher-order nonlinearity. The reductive perturbation method is used to obtain the second-order and third-order partial differential equations; then, Kodama and Taniuti's technique [J. Phys. Soc. Jpn. 45, 298 (1978)] is applied in order to remove the secularities in the third-order differential equations and derive a linear second-order inhomogeneous differential equation. The solution to this new second-ordermore » equation indicates that, as the amplitude increases, the hump-type Korteweg-de Vries solution is concentrated more around the center position of the soliton and that dip-type structures form near the two edges of the soliton. This result has a close relationship with the interpretation of the complex KASW structures observed in space with satellites.« less

Correcting the initialization of models with fractional derivatives via history-dependent conditions

NASA Astrophysics Data System (ADS)

Du, Maolin; Wang, Zaihua

2016-04-01

Fractional differential equations are more and more used in modeling memory (history-dependent, non-local, or hereditary) phenomena. Conventional initial values of fractional differential equations are defined at a point, while recent works define initial conditions over histories. We prove that the conventional initialization of fractional differential equations with a Riemann-Liouville derivative is wrong with a simple counter-example. The initial values were assumed to be arbitrarily given for a typical fractional differential equation, but we find one of these values can only be zero. We show that fractional differential equations are of infinite dimensions, and the initial conditions, initial histories, are defined as functions over intervals. We obtain the equivalent integral equation for Caputo case. With a simple fractional model of materials, we illustrate that the recovery behavior is correct with the initial creep history, but is wrong with initial values at the starting point of the recovery. We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.

1/f Noise from nonlinear stochastic differential equations.

PubMed

Ruseckas, J; Kaulakys, B

2010-03-01

We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of 1/fbeta noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating 1/fbeta noise, and provides further insights into the origin of 1/fbeta noise.

ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

NASA Astrophysics Data System (ADS)

Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

2018-04-01

In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

Improved accuracy of solar energy system testing and measurements

NASA Astrophysics Data System (ADS)

Waterman, R. E.

1984-12-01

A real world example is provided of recovery of data on the performance of a solar collector system in the field. Kalman filters were devised to reconstruct data from sensors which had functioned only intermittently over the 3-day trial period designed to quantify phenomena in the collector loop, i.e., hot water delivered to storage. The filter was configured to account for errors in data on the heat exchanger coil differential temperature and mass flow rate. Data were then generated based on a matrix of state equations, taking into account the presence of time delays due to tank stratification and convective flows. Good correlations were obtained with data from other sensors for the flow rate, system temperatures and the energy delivered to storage.

Machining Chatter Analysis for High Speed Milling Operations

NASA Astrophysics Data System (ADS)

Sekar, M.; Kantharaj, I.; Amit Siddhappa, Savale

2017-10-01

Chatter in high speed milling is characterized by time delay differential equations (DDE). Since closed form solution exists only for simple cases, the governing non-linear DDEs of chatter problems are solved by various numerical methods. Custom codes to solve DDEs are tedious to build, implement and not error free and robust. On the other hand, software packages provide solution to DDEs, however they are not straight forward to implement. In this paper an easy way to solve DDE of chatter in milling is proposed and implemented with MATLAB. Time domain solution permits the study and model of non-linear effects of chatter vibration with ease. Time domain results are presented for various stable and unstable conditions of cut and compared with stability lobe diagrams.

Micro-shimmy of towed structures in experimentally uncharted unstable parameter domain

NASA Astrophysics Data System (ADS)

Takács, Dénes; Stépán, Gábor

2012-11-01

In this paper, the lateral instability of towed structures (trailers, caravans and articulated buses) is investigated with special attention to the small amplitude lateral vibration that leads to a higher energy consumption in certain parameter domains. A low degree-of-freedom mechanical model of a shimmying towed tyre is used that describes the dynamics of the tyre-ground contact patch by the time delayed differential equation. Stability charts are calculated and the theoretically predicted linear unstable islands of small amplitude shimmy motions are validated by laboratory experiments. A tyre is towed by a relatively long caster, and its temperature and the input current of the conveyor belt are measured in order to show the increased value of the rolling resistance.

Auto-Bäcklund transformations for a matrix partial differential equation

NASA Astrophysics Data System (ADS)

Gordoa, P. R.; Pickering, A.

2018-07-01

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.

Long-Term Dynamics of Autonomous Fractional Differential Equations

NASA Astrophysics Data System (ADS)

Liu, Tao; Xu, Wei; Xu, Yong; Han, Qun

This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. The long-term dynamic behaviors predict where systems are heading after long-term evolution. We make some modification and transplant cell mapping methods to autonomous fractional differential equations. The mapping time duration of cell mapping is enlarged to deal with the long memory effect. Three illustrative examples, i.e. fractional Lotka-Volterra equation, fractional van der Pol oscillator and fractional Duffing equation, are studied with our revised generalized cell mapping method. We obtain long-term dynamics, such as attractors, basins of attraction, and saddles. Compared with some existing stability and numerical results, the validity of our method is verified. Furthermore, we find that the fractional order has its effect on the long-term dynamics of autonomous fractional differential equations.

Construction and accuracy of partial differential equation approximations to the chemical master equation.

PubMed

Grima, Ramon

2011-11-01

The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.

Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov-Kuznetsov equations

NASA Astrophysics Data System (ADS)

Huang, Ding-jiang; Ivanova, Nataliya M.

2016-02-01

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov-Kuznetsov (GZK) equations of form ut +(F (u)) xxx +(G (u)) xyy +(H (u)) x = 0. As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov-Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs.

Some problems in fractal differential equations

NASA Astrophysics Data System (ADS)

Su, Weiyi

2016-06-01

Based upon the fractal calculus on local fields, or p-type calculus, or Gibbs-Butzer calculus ([1],[2]), we suggest a constructive idea for "fractal differential equations", beginning from some special examples to a general theory. However, this is just an original idea, it needs lots of later work to support. In [3], we show example "two dimension wave equations with fractal boundaries", and in this note, other examples, as well as an idea to construct fractal differential equations are shown.

Parameter Estimates in Differential Equation Models for Chemical Kinetics

ERIC Educational Resources Information Center

Winkel, Brian

2011-01-01

We discuss the need for devoting time in differential equations courses to modelling and the completion of the modelling process with efforts to estimate the parameters in the models using data. We estimate the parameters present in several differential equation models of chemical reactions of order n, where n = 0, 1, 2, and apply more general…

Factors Affecting Differential Equation Problem Solving Ability of Students at Pre-University Level: A Conceptual Model

ERIC Educational Resources Information Center

Aisha, Bibi; Zamri, Sharifa NorulAkmar Syed; Abdallah, Nabeel; Abedalaziz, Mohammad; Ahmad, Mushtaq; Satti, Umbreen

2017-01-01

In this study, different factors affecting students' differential equations (DEs) solving abilities were explored at pre university level. To explore main factors affecting students' differential equations problem solving ability, articles for a 19-year period, from 1996 to 2015, were critically reviewed and analyzed. It was revealed that…

Nonstandard Topics for Student Presentations in Differential Equations

ERIC Educational Resources Information Center

LeMasurier, Michelle

2006-01-01

An interesting and effective way to showcase the wide variety of fields to which differential equations can be applied is to have students give short oral presentations on a specific application. These talks, which have been presented by 30-40 students per year in our differential equations classes, provide exposure to a diverse array of topics…

Existence of anti-periodic (differentiable) mild solutions to semilinear differential equations with nondense domain.

PubMed

Liu, Jinghuai; Zhang, Litao

2016-01-01

In this paper, we investigate the existence of anti-periodic (or anti-periodic differentiable) mild solutions to the semilinear differential equation [Formula: see text] with nondense domain. Furthermore, an example is given to illustrate our results.

Laplace and the era of differential equations

NASA Astrophysics Data System (ADS)

Weinberger, Peter

2012-11-01

Between about 1790 and 1850 French mathematicians dominated not only mathematics, but also all other sciences. The belief that a particular physical phenomenon has to correspond to a single differential equation originates from the enormous influence Laplace and his contemporary compatriots had in all European learned circles. It will be shown that at the beginning of the nineteenth century Newton's "fluxionary calculus" finally gave way to a French-type notation of handling differential equations. A heated dispute in the Philosophical Magazine between Challis, Airy and Stokes, all three of them famous Cambridge professors of mathematics, then serves to illustrate the era of differential equations. A remark about Schrödinger and his equation for the hydrogen atom finally will lead back to present times.

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

NASA Astrophysics Data System (ADS)

Ye, H.; Liu, F.; Turner, I.; Anh, V.; Burrage, K.

2013-09-01

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0, m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.

All-optical computation system for solving differential equations based on optical intensity differentiator.

PubMed

Tan, Sisi; Wu, Zhao; Lei, Lei; Hu, Shoujin; Dong, Jianji; Zhang, Xinliang

2013-03-25

We propose and experimentally demonstrate an all-optical differentiator-based computation system used for solving constant-coefficient first-order linear ordinary differential equations. It consists of an all-optical intensity differentiator and a wavelength converter, both based on a semiconductor optical amplifier (SOA) and an optical filter (OF). The equation is solved for various values of the constant-coefficient and two considered input waveforms, namely, super-Gaussian and Gaussian signals. An excellent agreement between the numerical simulation and the experimental results is obtained.

Theory of biaxial graded-index optical fiber. M.S. Thesis

NASA Technical Reports Server (NTRS)

Kawalko, Stephen F.

1990-01-01

A biaxial graded-index fiber with a homogeneous cladding is studied. Two methods, wave equation and matrix differential equation, of formulating the problem and their respective solutions are discussed. For the wave equation formulation of the problem it is shown that for the case of a diagonal permittivity tensor the longitudinal electric and magnetic fields satisfy a pair of coupled second-order differential equations. Also, a generalized dispersion relation is derived in terms of the solutions for the longitudinal electric and magnetic fields. For the case of a step-index fiber, either isotropic or uniaxial, these differential equations can be solved exactly in terms of Bessel functions. For the cases of an istropic graded-index and a uniaxial graded-index fiber, a solution using the Wentzel, Krammers and Brillouin (WKB) approximation technique is shown. Results for some particular permittivity profiles are presented. Also the WKB solutions is compared with the vector solution found by Kurtz and Streifer. For the matrix formulation it is shown that the tangential components of the electric and magnetic fields satisfy a system of four first-order differential equations which can be conveniently written in matrix form. For the special case of meridional modes, the system of equations splits into two systems of two equations. A general iterative technique, asymptotic partitioning of systems of equations, for solving systems of differential equations is presented. As a simple example, Bessel's differential equation is written in matrix form and is solved using this asymptotic technique. Low order solutions for particular examples of a biaxial and uniaxial graded-index fiber are presented. Finally numerical results obtained using the asymptotic technique are presented for particular examples of isotropic and uniaxial step-index fibers and isotropic, uniaxial and biaxial graded-index fibers.

Population stochastic modelling (PSM)--an R package for mixed-effects models based on stochastic differential equations.

PubMed

Klim, Søren; Mortensen, Stig Bousgaard; Kristensen, Niels Rode; Overgaard, Rune Viig; Madsen, Henrik

2009-06-01

The extension from ordinary to stochastic differential equations (SDEs) in pharmacokinetic and pharmacodynamic (PK/PD) modelling is an emerging field and has been motivated in a number of articles [N.R. Kristensen, H. Madsen, S.H. Ingwersen, Using stochastic differential equations for PK/PD model development, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 109-141; C.W. Tornøe, R.V. Overgaard, H. Agersø, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res. 22 (August(8)) (2005) 1247-1258; R.V. Overgaard, N. Jonsson, C.W. Tornøe, H. Madsen, Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 85-107; U. Picchini, S. Ditlevsen, A. De Gaetano, Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics, Math. Med. Biol. 25 (June(2)) (2008) 141-155]. PK/PD models are traditionally based ordinary differential equations (ODEs) with an observation link that incorporates noise. This state-space formulation only allows for observation noise and not for system noise. Extending to SDEs allows for a Wiener noise component in the system equations. This additional noise component enables handling of autocorrelated residuals originating from natural variation or systematic model error. Autocorrelated residuals are often partly ignored in PK/PD modelling although violating the hypothesis for many standard statistical tests. This article presents a package for the statistical program R that is able to handle SDEs in a mixed-effects setting. The estimation method implemented is the FOCE(1) approximation to the population likelihood which is generated from the individual likelihoods that are approximated using the Extended Kalman Filter's one-step predictions.

Hybrid modeling in biochemical systems theory by means of functional petri nets.

PubMed

Wu, Jialiang; Voit, Eberhard

2009-02-01

Many biological systems are genuinely hybrids consisting of interacting discrete and continuous components and processes that often operate at different time scales. It is therefore desirable to create modeling frameworks capable of combining differently structured processes and permitting their analysis over multiple time horizons. During the past 40 years, Biochemical Systems Theory (BST) has been a very successful approach to elucidating metabolic, gene regulatory, and signaling systems. However, its foundation in ordinary differential equations has precluded BST from directly addressing problems containing switches, delays, and stochastic effects. In this study, we extend BST to hybrid modeling within the framework of Hybrid Functional Petri Nets (HFPN). First, we show how the canonical GMA and S-system models in BST can be directly implemented in a standard Petri Net framework. In a second step we demonstrate how to account for different types of time delays as well as for discrete, stochastic, and switching effects. Using representative test cases, we validate the hybrid modeling approach through comparative analyses and simulations with other approaches and highlight the feasibility, quality, and efficiency of the hybrid method.

Understanding viral video dynamics through an epidemic modelling approach

NASA Astrophysics Data System (ADS)

Sachak-Patwa, Rahil; Fadai, Nabil T.; Van Gorder, Robert A.

2018-07-01

Motivated by the hypothesis that the spread of viral videos is analogous to the spread of a disease epidemic, we formulate a novel susceptible-exposed-infected-recovered-susceptible (SEIRS) delay differential equation epidemic model to describe the popularity evolution of viral videos. Our models incorporate time-delay, in order to accurately describe the virtual contact process between individuals and the temporary immunity of individuals to videos after they have grown tired of watching them. We validate our models by fitting model parameters to viewing data from YouTube music videos, in order to demonstrate that the model solutions accurately reproduce real behaviour seen in this data. We use an SEIR model to describe the initial growth and decline of daily views, and an SEIRS model to describe the long term behaviour of the popularity of music videos. We also analyse the decay rates in the daily views of videos, determining whether they follow a power law or exponential distribution. Although we focus on viral videos, the modelling approach may be used to understand dynamics emergent from other areas of science which aim to describe consumer behaviour.

Solving Nonlinear Coupled Differential Equations

NASA Technical Reports Server (NTRS)

Mitchell, L.; David, J.

1986-01-01

Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.

The Pendulum and the Calculus.

ERIC Educational Resources Information Center

Sworder, Steven C.

A pair of experiments, appropriate for the lower division fourth semester calculus or differential equations course, are presented. The second order differential equation representing the equation of motion of a simple pendulum is derived. The period of oscillation for a particular pendulum can be predicted from the solution to this equation. As a…

Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

Similarity solutions of reaction–diffusion equation with space- and time-dependent diffusion and reaction terms

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ho, C.-L.; Lee, C.-C., E-mail: chieh.no27@gmail.com

2016-01-15

We consider solvability of the generalized reaction–diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction–diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction–diffusion systems. Several representative examples of exactly solvable reaction–diffusion equations are presented.

Algebraic and geometric structures of analytic partial differential equations

NASA Astrophysics Data System (ADS)

Kaptsov, O. V.

2016-11-01

We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.

Symmetries of the Gas Dynamics Equations using the Differential Form Method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ramsey, Scott D.; Baty, Roy S.

Here, a brief review of the theory of exterior differential systems and isovector symmetry analysis methods is presented in the context of the one-dimensional inviscid compressible flow equations. These equations are formulated as an exterior differential system with equation of state (EOS) closure provided in terms of an adiabatic bulk modulus. The scaling symmetry generators—and corresponding EOS constraints—otherwise appearing in the existing literature are recovered through the application and invariance under Lie derivative dragging operations.

Symmetries of the Gas Dynamics Equations using the Differential Form Method

DOE PAGES

Ramsey, Scott D.; Baty, Roy S.

2017-11-21

Here, a brief review of the theory of exterior differential systems and isovector symmetry analysis methods is presented in the context of the one-dimensional inviscid compressible flow equations. These equations are formulated as an exterior differential system with equation of state (EOS) closure provided in terms of an adiabatic bulk modulus. The scaling symmetry generators—and corresponding EOS constraints—otherwise appearing in the existing literature are recovered through the application and invariance under Lie derivative dragging operations.

Modeling Individual Damped Linear Oscillator Processes with Differential Equations: Using Surrogate Data Analysis to Estimate the Smoothing Parameter

ERIC Educational Resources Information Center

Deboeck, Pascal R.; Boker, Steven M.; Bergeman, C. S.

2008-01-01

Among the many methods available for modeling intraindividual time series, differential equation modeling has several advantages that make it promising for applications to psychological data. One interesting differential equation model is that of the damped linear oscillator (DLO), which can be used to model variables that have a tendency to…

Improving Teaching Quality and Problem Solving Ability through Contextual Teaching and Learning in Differential Equations: A Lesson Study Approach

ERIC Educational Resources Information Center

Khotimah, Rita Pramujiyanti; Masduki

2016-01-01

Differential equations is a branch of mathematics which is closely related to mathematical modeling that arises in real-world problems. Problem solving ability is an essential component to solve contextual problem of differential equations properly. The purposes of this study are to describe contextual teaching and learning (CTL) model in…

Differential geometry techniques for sets of nonlinear partial differential equations

NASA Technical Reports Server (NTRS)

Estabrook, Frank B.

1990-01-01

An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.

On the Singular Perturbations for Fractional Differential Equation

PubMed Central

Atangana, Abdon

2014-01-01

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method. PMID:24683357

A neuro approach to solve fuzzy Riccati differential equations

NASA Astrophysics Data System (ADS)

Shahrir, Mohammad Shazri; Kumaresan, N.; Kamali, M. Z. M.; Ratnavelu, Kurunathan

2015-10-01

There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2016-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

Solution of some types of differential equations: operational calculus and inverse differential operators.

PubMed

Zhukovsky, K

2014-01-01

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.

A neuro approach to solve fuzzy Riccati differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shahrir, Mohammad Shazri, E-mail: mshazri@gmail.com; Telekom Malaysia, R&D TM Innovation Centre, LingkaranTeknokrat Timur, 63000 Cyberjaya, Selangor; Kumaresan, N., E-mail: drnk2008@gmail.com

There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.

Dimensional analysis yields the general second-order differential equation underlying many natural phenomena: the mathematical properties of a phenomenon's data plot then specify a unique differential equation for it.

PubMed

Kepner, Gordon R

2014-08-27

This study uses dimensional analysis to derive the general second-order differential equation that underlies numerous physical and natural phenomena described by common mathematical functions. It eschews assumptions about empirical constants and mechanisms. It relies only on the data plot's mathematical properties to provide the conditions and constraints needed to specify a second-order differential equation that is free of empirical constants for each phenomenon. A practical example of each function is analyzed using the general form of the underlying differential equation and the observable unique mathematical properties of each data plot, including boundary conditions. This yields a differential equation that describes the relationship among the physical variables governing the phenomenon's behavior. Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach. It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.

Robustness analysis of an air heating plant and control law by using polynomial chaos

DOE Office of Scientific and Technical Information (OSTI.GOV)

Colón, Diego; Ferreira, Murillo A. S.; Bueno, Átila M.

2014-12-10

This paper presents a robustness analysis of an air heating plant with a multivariable closed-loop control law by using the polynomial chaos methodology (MPC). The plant consists of a PVC tube with a fan in the air input (that forces the air through the tube) and a mass flux sensor in the output. A heating resistance warms the air as it flows inside the tube, and a thermo-couple sensor measures the air temperature. The plant has thus two inputs (the fan's rotation intensity and heat generated by the resistance, both measured in percent of the maximum value) and two outputsmore » (air temperature and air mass flux, also in percent of the maximal value). The mathematical model is obtained by System Identification techniques. The mass flux sensor, which is nonlinear, is linearized and the delays in the transfer functions are properly approximated by non-minimum phase transfer functions. The resulting model is transformed to a state-space model, which is used for control design purposes. The multivariable robust control design techniques used is the LQG/LTR, and the controllers are validated in simulation software and in the real plant. Finally, the MPC is applied by considering some of the system's parameters as random variables (one at a time, and the system's stochastic differential equations are solved by expanding the solution (a stochastic process) in an orthogonal basis of polynomial functions of the basic random variables. This method transforms the stochastic equations in a set of deterministic differential equations, which can be solved by traditional numerical methods (That is the MPC). Statistical data for the system (like expected values and variances) are then calculated. The effects of randomness in the parameters are evaluated in the open-loop and closed-loop pole's positions.« less

A three operator split-step method covering a larger set of non-linear partial differential equations

NASA Astrophysics Data System (ADS)

Zia, Haider

2017-06-01

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.

On the solution of the generalized wave and generalized sine-Gordon equations

NASA Technical Reports Server (NTRS)

Ablowitz, M. J.; Beals, R.; Tenenblat, K.

1986-01-01

The generalized wave equation and generalized sine-Gordon equations are known to be natural multidimensional differential geometric generalizations of the classical two-dimensional versions. In this paper, a system of linear differential equations is associated with these equations, and it is shown how the direct and inverse problems can be solved for appropriately decaying data on suitable lines. An initial-boundary value problem is solved for these equations.

Cellular Automata for Spatiotemporal Pattern Formation from Reaction-Diffusion Partial Differential Equations

NASA Astrophysics Data System (ADS)

Ohmori, Shousuke; Yamazaki, Yoshihiro

2016-01-01

Ultradiscrete equations are derived from a set of reaction-diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction-diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction-diffusion equations.

Lie symmetries and conservation laws for the time fractional Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

Rui, Wenjuan; Zhang, Xiangzhi

2016-05-01

This paper investigates the invariance properties of the time fractional Derrida-Lebowitz-Speer-Spohn (FDLSS) equation with Riemann-Liouville derivative. By using the Lie group analysis method of fractional differential equations, we derive Lie symmetries for the FDLSS equation. In a particular case of scaling transformations, we transform the FDLSS equation into a nonlinear ordinary fractional differential equation. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

Solving constant-coefficient differential equations with dielectric metamaterials

NASA Astrophysics Data System (ADS)

Zhang, Weixuan; Qu, Che; Zhang, Xiangdong

2016-07-01

Recently, the concept of metamaterial analog computing has been proposed (Silva et al 2014 Science 343 160-3). Some mathematical operations such as spatial differentiation, integration, and convolution, have been performed by using designed metamaterial blocks. Motivated by this work, we propose a practical approach based on dielectric metamaterial to solve differential equations. The ordinary differential equation can be solved accurately by the correctly designed metamaterial system. The numerical simulations using well-established numerical routines have been performed to successfully verify all theoretical analyses.

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression.

PubMed

Ding, A Adam; Wu, Hulin

2014-10-01

We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method.

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression

PubMed Central

Ding, A. Adam; Wu, Hulin

2015-01-01

We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method. PMID:26401093

Modelling with Difference Equations Supported by GeoGebra: Exploring the Kepler Problem

ERIC Educational Resources Information Center

Kovacs, Zoltan

2010-01-01

The use of difference and differential equations in the modelling is a topic usually studied by advanced students in mathematics. However difference and differential equations appear in the school curriculum in many direct or hidden ways. Difference equations first enter in the curriculum when studying arithmetic sequences. Moreover Newtonian…

Alternate Solution to Generalized Bernoulli Equations via an Integrating Factor: An Exact Differential Equation Approach

ERIC Educational Resources Information Center

Tisdell, C. C.

2017-01-01

Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…

Intuitive Understanding of Solutions of Partially Differential Equations

ERIC Educational Resources Information Center

Kobayashi, Y.

2008-01-01

This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based…

Pulse-coupled mixed-mode oscillators: Cluster states and extreme noise sensitivity

NASA Astrophysics Data System (ADS)

Karamchandani, Avinash J.; Graham, James N.; Riecke, Hermann

2018-04-01

Motivated by rhythms in the olfactory system of the brain, we investigate the synchronization of all-to-all pulse-coupled neuronal oscillators exhibiting various types of mixed-mode oscillations (MMOs) composed of sub-threshold oscillations (STOs) and action potentials ("spikes"). We focus particularly on the impact of the delay in the interaction. In the weak-coupling regime, we reduce the system to a Kuramoto-type equation with non-sinusoidal phase coupling and the associated Fokker-Planck equation. Its linear stability analysis identifies the appearance of various cluster states. Their type depends sensitively on the delay and the width of the pulses. Interestingly, long delays do not imply slow population rhythms, and the number of emerging clusters only loosely depends on the number of STOs. Direct simulations of the oscillator equations reveal that for quantitative agreement of the weak-coupling theory the coupling strength and the noise have to be extremely small. Even moderate noise leads to significant skipping of STO cycles, which can enhance the diffusion coefficient in the Fokker-Planck equation by two orders of magnitude. Introducing an effective diffusion coefficient extends the range of agreement significantly. Numerical simulations of the Fokker-Planck equation reveal bistability and solutions with oscillatory order parameters that result from nonlinear mode interactions. These are confirmed in simulations of the full spiking model.

Stability for a class of difference equations

NASA Astrophysics Data System (ADS)

Muroya, Yoshiaki; Ishiwata, Emiko

2009-06-01

We consider the following non-autonomous and nonlinear difference equations with unbounded delays: where 0

Delayed-enhanced cardiac MRI for differentiation of Fabry's disease from symmetric hypertrophic cardiomyopathy.

PubMed

De Cobelli, Francesco; Esposito, Antonio; Belloni, Elena; Pieroni, Maurizio; Perseghin, Gianluca; Chimenti, Cristina; Frustaci, Andrea; Del Maschio, Alessandro

2009-03-01

Fabry's disease may be difficult to differentiate from symmetric hypertrophic cardiomyopathy. Our aim was to compare the myocardial location and distribution patterns of delayed enhancement between patients with Fabry's disease who are affected by symmetric myocardial hypertrophy and patients with symmetric hypertrophic cardiomyopathy in order to identify a specific sign to best differentiate the two diseases. Patients with Fabry's disease-related hypertrophy showed left ventricular (LV) delayed enhancement with a typical and consistently found pattern characterized by the involvement of the inferolateral basal or mid basal segments and a mesocardial distribution that spared the subendocardium. This pattern seems to be specific to Fabry's disease; in fact, patients with symmetric hypertrophic cardiomyopathy had variable locations and distributions of delayed enhancement. These observations may contribute to identifying Fabry's disease as a specific cause of symmetric hypertrophy.

Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

PubMed Central

Müller, Eike H.; Scheichl, Rob; Shardlow, Tony

2015-01-01

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy. PMID:27547075

Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation.

PubMed

Müller, Eike H; Scheichl, Rob; Shardlow, Tony

2015-04-08

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

Differential Equation Models for Sharp Threshold Dynamics

DTIC Science & Technology

2012-08-01

dynamics, and the Lanchester model of armed conflict, where the loss of a key capability drastically changes dynamics. We derive and demonstrate a step...dynamics using differential equations. 15. SUBJECT TERMS Differential Equations, Markov Population Process, S-I-R Epidemic, Lanchester Model 16...infection, where a detection event drastically changes dynamics, and the Lanchester model of armed conflict, where the loss of a key capability

A Simple Method to Find out when an Ordinary Differential Equation Is Separable

ERIC Educational Resources Information Center

Cid, Jose Angel

2009-01-01

We present an alternative method to that of Scott (D. Scott, "When is an ordinary differential equation separable?", "Amer. Math. Monthly" 92 (1985), pp. 422-423) to teach the students how to discover whether a differential equation y[prime] = f(x,y) is separable or not when the nonlinearity f(x, y) is not explicitly factorized. Our approach is…

Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

PubMed Central

Ahmad, M. Z.; Hasan, M. K.; Abbasbandy, S.

2013-01-01

We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided. PMID:24082853

Discovery and Optimization of Low-Storage Runge-Kutta Methods

DTIC Science & Technology

2015-06-01

NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS DISCOVERY AND OPTIMIZATION OF LOW-STORAGE RUNGE-KUTTA METHODS by Matthew T. Fletcher June 2015... methods are an important family of iterative methods for approximating the solutions of ordinary differential equations (ODEs) and differential...algebraic equations (DAEs). It is common to use an RK method to discretize in time when solving time dependent partial differential equations (PDEs) with a

Asymptotic analysis of the local potential approximation to the Wetterich equation

NASA Astrophysics Data System (ADS)

Bender, Carl M.; Sarkar, Sarben

2018-06-01

This paper reports a study of the nonlinear partial differential equation that arises in the local potential approximation to the Wetterich formulation of the functional renormalization group equation. A cut-off-dependent shift of the potential in this partial differential equation is performed. This shift allows a perturbative asymptotic treatment of the differential equation for large values of the infrared cut-off. To leading order in perturbation theory the differential equation becomes a heat equation, where the sign of the diffusion constant changes as the space-time dimension D passes through 2. When D  <  2, one obtains a forward heat equation whose initial-value problem is well-posed. However, for D  >  2 one obtains a backward heat equation whose initial-value problem is ill-posed. For the special case D  =  1 the asymptotic series for cubic and quartic models is extrapolated to the small infrared-cut-off limit by using Padé techniques. The effective potential thus obtained from the partial differential equation is then used in a Schrödinger-equation setting to study the stability of the ground state. For cubic potentials it is found that this Padé procedure distinguishes between a -symmetric theory and a conventional Hermitian theory (g real). For an theory the effective potential is nonsingular and has a stable ground state but for a conventional theory the effective potential is singular. For a conventional Hermitian theory and a -symmetric theory (g  >  0) the results are similar; the effective potentials in both cases are nonsingular and possess stable ground states.

Distributed Optimal Consensus Control for Multiagent Systems With Input Delay.

PubMed

Zhang, Huaipin; Yue, Dong; Zhao, Wei; Hu, Songlin; Dou, Chunxia; Huaipin Zhang; Dong Yue; Wei Zhao; Songlin Hu; Chunxia Dou; Hu, Songlin; Zhang, Huaipin; Dou, Chunxia; Yue, Dong; Zhao, Wei

2018-06-01

This paper addresses the problem of distributed optimal consensus control for a continuous-time heterogeneous linear multiagent system subject to time varying input delays. First, by discretization and model transformation, the continuous-time input-delayed system is converted into a discrete-time delay-free system. Two delicate performance index functions are defined for these two systems. It is shown that the performance index functions are equivalent and the optimal consensus control problem of the input-delayed system can be cast into that of the delay-free system. Second, by virtue of the Hamilton-Jacobi-Bellman (HJB) equations, an optimal control policy for each agent is designed based on the delay-free system and a novel value iteration algorithm is proposed to learn the solutions to the HJB equations online. The proposed adaptive dynamic programming algorithm is implemented on the basis of a critic-action neural network (NN) structure. Third, it is proved that local consensus errors of the two systems and weight estimation errors of the critic-action NNs are uniformly ultimately bounded while the approximated control policies converge to their target values. Finally, two simulation examples are presented to illustrate the effectiveness of the developed method.

Group foliation of finite difference equations

NASA Astrophysics Data System (ADS)

Thompson, Robert; Valiquette, Francis

2018-06-01

Using the theory of equivariant moving frames, a group foliation method for invariant finite difference equations is developed. This method is analogous to the group foliation of differential equations and uses the symmetry group of the equation to decompose the solution process into two steps, called resolving and reconstruction. Our constructions are performed algorithmically and symbolically by making use of discrete recurrence relations among joint invariants. Applications to invariant finite difference equations that approximate differential equations are given.

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4).

PubMed

Murase, Kenya

2016-01-01

Partial differential equations are often used in the field of medical physics. In this (final) issue, the methods for solving the partial differential equations were introduced, which include separation of variables, integral transform (Fourier and Fourier-sine transforms), Green's function, and series expansion methods. Some examples were also introduced, in which the integral transform and Green's function methods were applied to solving Pennes' bioheat transfer equation and the Fourier series expansion method was applied to Navier-Stokes equation for analyzing the wall shear stress in blood vessels.Finally, the author hopes that this series will be helpful for people who engage in medical physics.

Stability of Detached Solidification

NASA Technical Reports Server (NTRS)

Mazuruk, K.; Volz, M. P.; Croell, A.

2009-01-01

Bridgman crystal growth can be conducted in the so-called "detached" solidification regime, where the growing crystal is detached from the crucible wall. A small gap between the growing crystal and the crucible wall, of the order of 100 micrometers or less, can be maintained during the process. A meniscus is formed at the bottom of the melt between the crystal and crucible wall. Under proper conditions, growth can proceed without collapsing the meniscus. The meniscus shape plays a key role in stabilizing the process. Thermal and other process parameters can also affect the geometrical steady-state stability conditions of solidification. The dynamic stability theory of the shaped crystal growth process has been developed by Tatarchenko. It consists of finding a simplified autonomous set of differential equations for the radius, height, and possibly other process parameters. The problem then reduces to analyzing a system of first order linear differential equations for stability. Here we apply a modified version of this theory for a particular case of detached solidification. Approximate analytical formulas as well as accurate numerical values for the capillary stability coefficients are presented. They display an unexpected singularity as a function of pressure differential. A novel approach to study the thermal field effects on the crystal shape stability has been proposed. In essence, it rectifies the unphysical assumption of the model that utilizes a perturbation of the crystal radius along the axis as being instantaneous. It consists of introducing time delay effects into the mathematical description and leads, in general, to stability over a broader parameter range. We believe that this novel treatment can be advantageously implemented in stability analyses of other crystal growth techniques such as Czochralski and float zone methods.

Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

NASA Astrophysics Data System (ADS)

Anosov, Dmitry V.; Leksin, Vladimir P.

2011-02-01

This paper contains an account of A.A. Bolibrukh's results obtained in the new directions of research that arose in the analytic theory of differential equations as a consequence of his sensational counterexample to the Riemann-Hilbert problem. A survey of results of his students in developing topics first considered by Bolibrukh is also presented. The main focus is on the role of the reducibility/irreducibility of systems of linear differential equations and their monodromy representations. A brief synopsis of results on the multidimensional Riemann-Hilbert problem and on isomonodromic deformations of Fuchsian systems is presented, and the main methods in the modern analytic theory of differential equations are sketched. Bibliography: 69 titles.

Green function of the double-fractional Fokker-Planck equation: path integral and stochastic differential equations.

PubMed

Kleinert, H; Zatloukal, V

2013-11-01

The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

A New Factorisation of a General Second Order Differential Equation

ERIC Educational Resources Information Center

Clegg, Janet

2006-01-01

A factorisation of a general second order ordinary differential equation is introduced from which the full solution to the equation can be obtained by performing two integrations. The method is compared with traditional methods for solving these type of equations. It is shown how the Green's function can be derived directly from the factorisation…

Quantum spatial propagation of squeezed light in a degenerate parametric amplifier

NASA Technical Reports Server (NTRS)

Deutsch, Ivan H.; Garrison, John C.

1992-01-01

Differential equations which describe the steady state spatial evolution of nonclassical light are established using standard quantum field theoretic techniques. A Schroedinger equation for the state vector of the optical field is derived using the quantum analog of the slowly varying envelope approximation (SVEA). The steady state solutions are those that satisfy the time independent Schroedinger equation. The resulting eigenvalue problem then leads to the spatial propagation equations. For the degenerate parametric amplifier this method shows that the squeezing parameter obey nonlinear differential equations coupled by the amplifier gain and phase mismatch. The solution to these differential equations is equivalent to one obtained from the classical three wave mixing steady state solution to the parametric amplifier with a nondepleted pump.

Dynamically orthogonal field equations for stochastic flows and particle dynamics

DTIC Science & Technology

2011-02-01

where uncertainty ‘lives’ as well as a system of Stochastic Di erential Equations that de nes how the uncertainty evolves in the time varying stochastic ... stochastic dynamical component that are both time and space dependent, we derive a system of field equations consisting of a Partial Differential Equation...a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new

Non-invertible transformations of differential-difference equations

NASA Astrophysics Data System (ADS)

Garifullin, R. N.; Yamilov, R. I.; Levi, D.

2016-09-01

We discuss aspects of the theory of non-invertible transformations of differential-difference equations and, in particular, the notion of Miura type transformation. We introduce the concept of non-Miura type linearizable transformation and we present techniques that allow one to construct simple linearizable transformations and might help one to solve classification problems. This theory is illustrated by the example of a new integrable differential-difference equation depending on five lattice points, interesting from the viewpoint of the non-invertible transformation, which relate it to an Itoh-Narita-Bogoyavlensky equation.

Dynamic characteristics of a two-stage variable-mass flexible missile with internal flow

NASA Technical Reports Server (NTRS)

Meirovitch, L.; Bankovskis, J.

1972-01-01

A general formulation of the dynamical problems associated with powered flight of a two stage flexible, variable-mass missile with internal flow, discrete masses, and aerodynamic forces is presented. The formulation comprises six ordinary differential equations for the rigid body motion, 3n ordinary differential equations for the n discrete masses and three partial differential equations with the appropriate boundary conditions for the elastic motion. This set of equations is modified to represent a single stage flexible, variable-mass missile with internal flow and aerodynamic forces. The rigid-body motion consists then of three translations and three rotations, whereas the elastic motion is defined by one longitudinal and two flexural displacements, the latter about two orthogonal transverse axes. The differential equations are nonlinear and, in addition, they possess time-dependent coefficients due to the mass variation.