Dynamics of constrained differential delay equations

2014

Several aspects of global dynamics and the existence of periodic solutions are studied for the scalar differential delay equation x ′(t) = a(t)f(x([t − K])), where f(x) is a continuous negative feedback function, x · f(x) < 0, x 6= 0, 0 ≤ a(t) is continuous ω-periodic, [·] is the integer part function, and the integer K ≥ 0 is the delay. The case of integer period ω allows for a reduction to finite-dimensional difference equations. The dynamics of the latter are studied in terms of corresponding discrete maps, including the partial case of interval maps (K = 0).

The analysis and numerical solution of initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Characteristic properties of DDAEs are analyzed and the differences between causal and noncausal DDAEs are studied. The method of steps is analyzed and it is shown that it has to be modified for general DDAEs. The classification of ordinary delay differential equations (DDEs) is generalized to DDAEs, and a numerical solution procedure for general retarded and neutral DDAEs is constructed. The properties of the algorithm are studied and the theoretical results are illustrated with a numerical example.

2019

We prove the existence of periodic solutions of the differential delay equation εx˙(t)+x(t)=f(x(t−1)),ε>0 under the assumptions that the continuous nonlinearity f(x) satisfies the negative feedback condition, x⋅f(x)<0,x≠0, has sufficiently large derivative at zero |f′(0)|, and possesses an invariant interval I∋0,f(I)⊆I, as a dimensional map. As ε→0+ we show the convergence of the periodic solutions to a discontinuous square wave function generated by the globally attracting 2-cycle of the map f.

2012

The pervading theme of this thesis is the development of insights that contribute to the understanding of whether certain classes of functional dif ferential equation have solutions that are all oscillatory. The starting point for the work is the analysis of simple (linear au tonomous) ordinary differential equations where existing results allow a full explanation of the phenomena. The Laplace transform features as a key tool in developing a theoretical background. The thesis goes on to explore the corresponding theory for delay equa tions, advanced equations and functional differential equations of mixed type. The focus is on understanding the links between the characteristic roots of the underlying equation, and the presence or otherwise of oscillatory solu tions. The linear t?-methods are used as a class of numerical schemes which lead to discrete problems analogous to each of the classes of functional differential equation under consideration. The thesis goes on to discuss the insights that can be obtained for discrete problems in their own right, and then considers those new insights that can be obtained about the underlying continuous problem from analysis of the oscillatory behaviour of the analogous discrete problem. The main conclusions of the work are some semi-automated computa tional approaches (based upon the Principle of the Argument) which allow the prediction of oscillatory solutions to be made. Examples of the effec tiveness of the approach are provided, and there is some discussion of its theoretical basis. The thesis concludes with some observations about further work and some of the limitations of existing analytical insights which restrict the reliability with which the approach developed can be applied to wider classes of problem.

Journal of Differential Equations, 2006

We prove that a planar delay differential equation subjected to a feedback condition and a self-supporting impulsive condition has periodic solutions. It also can have backset solutions, that encircle the origin opposing the orientation induced by the feedback condition.

International Journal of Bifurcation and Chaos, 1997

We present a new numerical method for the efficient computation of periodic solutions of nonlinear systems of Delay Differential Equations (DDEs) with several discrete delays. This method exploits the typical spectral properties of the monodromy matrix of a DDE and allows effective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. We show that the method is particularly suited to trace a branch of periodic solutions using continuation and can be used to locate bifurcation points with good accuracy.

In this paper we consider scalar linear periodic delay differential equations of the form are continuous periodic functions with period . We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with x ed step-size is applied to approximate the solution to ( ) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

Nonlinear Dynamics

This article presents an extension of the Asymptotic Numerical Method combined with the Harmonic Balance Method to the continuation of periodic orbits of Delay Differential Equations. The equations can be forced or autonomous and possibly of neutral type. The approach developed in this paper requires the system of equations to be written in a quadratic formalism which is detailed. The method is applied to various systems, from Van der Pol and Duffing oscillators to toy models of clarinet and saxophone. The Harmonic Balance Method is ascertained from a comparison to standards time-integration solvers. Bifurcation diagrams are drawn which are sometimes intricate, showing the robustness of this method.

Dynamics of Continuous, Discrete & Impulsive Systems. Series A, 2008

MU Akhmeta, JO Alzabutb and A. Zafera aDepartment of Mathematics, Middle East Technical University 06531 Ankara, Turkey E-mail:[email protected]; [email protected] bDepartment of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey E-mail: [email protected] ... Abstract. A necessary and sufficient condition is established for the existence of periodic solutions of linear impulsive delay differential systems. Keywords. Linear, Impulse, Delay, Adjoint, Periodic solution. AMS (MOS) subject classification: 34A37.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019

Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue ‘Nonlinear dynamics of delay systems’, which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter. This article is part of the theme issue ‘Nonlinear dynamics of delay...

2018

Delay differential equations have a wide range of applications in engineering. This work is devoted to the analysis of delay Duffing equation, which plays a crucial role in modeling performance on demand Micro Electro Mechanical Systems (MEMS). We start with the stability analysis of a linear delay model. We also show that in certain cases the delay model can be efficiently approximated with a much simpler model without delay. We proceed with the analysis of a non-linear Duffing equation. This model is a significantly more complex mathematical model. For instance, the existence of a periodic solution for this equation is a highly nontrivial question, which was established by Struwe [3]. The main result of this work is to establish the existence of a periodic solution to delay Duffing equation. The paper [7] claimed to establish the existence of such solutions, however their argument is wrong. In this work we establish the existence of a periodic solution under the assumption that th...

Nonlinear Physical Science, 2011

Periodic motions in DDE (Differential-Delay Equations) are typically created in Hopf bifurcations. In this chapter we examine this process from several points of view. Firstly we use Lindstedt's perturbation method to derive the Hopf Bifurcation Formula, which determines the stability of the periodic motion. Then we use the Two Variable Expansion Method (also known as Multiple Scales) to investigate the transient behavior involved in the approach to the periodic motion. Next we use Center Manifold Analysis to reduce the DDE from an an infinite dimensional evolution equation on a function space to a two dimensional ODE (Ordinary Differential Equation) on the center manifold, the latter being a surface tangent to the eigenspace associated with the Hopf bifurcation. Finally we provide an application to gene copying in which the delay is due to an observed time lag in the transcription process.

In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

Open Mathematics

In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x″(t) + px″(t − 1) = qx([t]) + f(t), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f(t) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.

Ann Arbor MI, 2003

A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear delay differential equations using the matrix form of DDEs. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Stability criteria for the individual modes, free response, and forced response for delay equations in different examples are studied, and the results are presented. The new approach is applied to obtain the stability regions for the individual modes of the linearized chatter problem in turning. The results present a necessary condition to the stability in chatter for the whole system, since it only enables the study of the individual modes, and there are an infinite number of them that contribute to the stability of the system.