Retarded Functional Differential Equations Research Articles

A Degenerate Hopf Bifurcation in Retarded Functional Differential Equations, and Applications to Endemic Bubbles

In this paper, we study degenerate Hopf bifurcations in a class of parametrized retarded functional differential equations. Specifically, we are interested in the case where the eigenvalue crossing condition of the classical Hopf bifurcation theorem is violated. Our approach is based on center manifold reduction and Poincare-Birkhoff normal forms, and a singularity theoretical classification of this degenerate Hopf bifurcation. Our results are then applied to a recently developed SIS model incorporating a delayed behavioral response. We show that the phenomenon of endemic bubbles, which is characterized by a branch of periodic solutions which bifurcates from the endemic equilibrium at some value of the basic reproduction number R0, and then reconnects to the endemic equilibrium at a larger value of R0, originates in a codimension-two organizing center where the eigenvalue crossing condition for the Hopf bifurcation theorem is violated.

Optimal Control Problems for Semilinear Retarded Functional Differential Equations

This paper deals with the time optimal control problem for semilinear retarded functional differential equations by using the construction of the fundamental solution in the case where the principal operators are unbounded. We also present the maximum principle for controls which are described by the adjoint state corresponding to the linear equations without the condition of differentiability for the nonlinear term.

Almost periodic solutions of partial differential equations with delay

In this paper we study the existence of almost periodic solutions for linear retarded functional differential equations with finite delay and values in a Banach space. We relate the existence of almost periodic solutions with the stabilization of distributed control systems. We apply our results to transport models and to the wave equation.

Asymptotic pattern for a partial neutral functional differential equation

In the present article, the asymptotic speed of spread and traveling wavefronts of a partial neutral functional differential equation (PNFDE) are investigated. A variable transform yields to a partial retarded functional differential equation (PRFDE) with infinite number of constant delays. The relation between the above PNFDE and PRFDE is studied and an equivalence in a certain meaning of these two equations is verified. Using this equivalence, we in fact obtain the existence and uniqueness of solution for the initial value problem of the PNFDE. We then consider the truncated system of the above PRFDE, which is a partial retarded functional differential equation with finite number of constant delays. We establish the existence of the traveling wave solution and the spreading speed for this truncated system. A limit argument and the method of finite delay approximation are used to obtain the existence of the traveling wave solutions and the spreading speed for the PRFDE with infinite number of constant delays. This leads to the existence of the traveling wave solution and the spreading speed for the considered PNFDE by the above equivalence. At last, we confirm that the spreading speed is coincidental with the minimal wave speed for all equations. An example is given to illustrate the application of our results.

Extended Hopf Bifurcation for Abstract Integral Equations at Resonant Eigenvalue

In this paper, we apply an extension of the abstract Hopf bifurcation theorem stated in Jacimovic (Non Anal TMA 73(8):2426---2432, 2010) to abstract integral equations (AIE) and retarded functional differential equations (RFDE). This yields sufficient conditions of what we refer to as extended Hopf bifurcation for AIE and RFDE, in which we have a relaxation of the non-resonance condition on the eigenvalues of the generator of corresponding semigroup. We illustrate our results with an explicit example of a system of two delay differential equations (DDE), undergoing extended Hopf bifurcation at the resonant eigenvalue.

Good Behavior with Respect to the Stiffness in the Numerical Integration of Retarded Functional Differential Equations

In this paper we obtain, for the global errors of a functional continuous Runge--Kutta (FCRK) method as applied to a retarded functional differential equation (RFDE), a recursive relation similar to that obtained for the global errors of a one-step method as applied to an ordinary differential equation. After which, we introduce a notion of good behavior with respect to the stiffness of an FCRK method on a given family of RFDEs. Finally, we analyze this notion of “good behavior” in the case of particular families of scalar semilinear RFDEs with nonvanishing delays.

One method for the investigation of linear functional-differential equations

We consider a scalar linear retarded functional-differential equation $$ \dot{x}(t)=ax\left( {t-1} \right)+bx\left( {\frac{t}{q}} \right)+f(t),\quad q>1. $$ The study of linear retarded functional-differential equations mainly deals with two types of initial-value problems: initial-value problem with initial functions and initial-value problems with initial point (when it is necessary to find a classical solution whose substitution in the original equation reduces it to the identity). In the paper, the initial-value problem with initial point is studied by the method of polynomial quasisolutions. We prove theorems on the existence of polynomial quasisolutions and exact polynomial solutions of the analyzed linear retarded functional-differential equation. The results of numerical experiments are presented.

Global continuation of periodic solutions for retarded functional differential equations on manifolds

Abstract We consider T-periodic parametrized retarded functional differential equations, with infinite delay, on (possibly) noncompact manifolds. Using a topological approach, based on the notions of degree of a tangent vector field and of the fixed point index, we prove a global continuation result for T-periodic solutions of such equations. Our main theorem is a generalization to the case of retarded equations of a global continuation result obtained by the last two authors for ordinary differential equations on manifolds. As corollaries we obtain a Rabinowitz-type global bifurcation result and a continuation principle of Mawhin type. MSC:34K13, 34C40, 37C25, 70K42.

Regularity for solutions of nonlinear retarded functional differential equations

This paper deals with the existence and uniqueness of solutions for the nonlinear func- tional differential equations with time delay. The regularity and a variation of constant formula for solutions of the given equations are also studied.

Stability property and essential spectrum of linear retarded functional differential equations

In this work, we shall study stability and stabilization of a class of retarded functional differential equations in Banach spaces. We present sufficient (and necessary, sometimes) conditions for weak, asymptotic and exponential stability properties. It is shown that retarded growth bounds are totally determined by retarded non-isolated spectra, isolated eigenvalues with infinite algebraic multiplicity and spectral bounds. Stabilization problem via compact perturbations, a topic which is practically useful in the context of control theory of dynamical systems, is considered as well.

Approximation of Eigenvalues of Evolution Operators for Linear Retarded Functional Differential Equations

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided.

Regularly Varying Solutions of Second Order Nonlinear Functional Differential Equations with Retarded Argument

The existence of slowly and regularly varying solutions in the sense of Karamata implying nonoscillation is proved for a class of second order nonlinear retarded functional differential equations of Thomas-Fermi type. A motivation for such study is the extensively developed theory offering a number of properties of regularly and slowly varying functions ([2]) - consequently of such solutions of differential equations. As an illustration, the precise asymptotic behaviour for $t\rightarrow \infty$ of the slowly varying solutions for a subclass of considered equations is presented.

Hopf-transcritical bifurcation in retarded functional differential equations

Firstly, we analyze a codimension-two unfolding for the Hopf-transcritical bifurcation, and give complete bifurcation diagrams and phase portraits. In particular, we express explicitly the heteroclinic bifurcation curve, and obtain conditions under which the secondary bifurcation periodic solutions and the heteroclinic orbit are stable. Secondly, we show how to reduce general retarded functional differential equation, with perturbation parameters near the critical point of the Hopf-transcritical bifurcation, to a 3-dimensional ordinary differential equation which is restricted on the center manifold up to the third order with unfolding parameters, and further reduce it to a 2-dimensional amplitude system, where these unfolding parameters can be expressed by those original perturbation parameters. Finally, we apply the general results to the van der Pol’s equation with delayed feedback, and obtain the existence of stable or unstable equilibria, periodic solutions and quasi-periodic solutions.

Equivariant normal forms for neutral functional differential equations

This paper addresses the computation of equivariant normal forms for some Neutral Functional Differential Equations (NFDEs) near equilibria in the presence of symmetry. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations (FDEs) and on the existence of center (or other invariant) manifolds. We illustrate our general results by some applications to a detailed case study of additive neurons with delayed feedback.

Global exponential stability of impulsive differential equations with any time delays

The main objective of this letter is to further investigate the global exponential stability of a class of general impulsive retarded functional differential equations. Several new criteria on global exponential stability are analytically established based on Lyapunov function methods combined with Razumikhin techniques. The obtained results extend and generalize some results existing in the literature. An example, along with computer simulations, is included to illustrate the results.

Numerical approximation of characteristic values of partial retarded functional differential equations

The stability of an equilibrium point of a dynamical system is determined by the position in the complex plane of the so-called characteristic values of the linearization around the equilibrium. This paper presents an approach for the computation of characteristic values of partial differential equations of evolution involving time delay, which is based on a pseudospectral method coupled with a spectral method. The convergence of the computed characteristic values is of infinite order with respect to the pseudospectral discretization and of finite order with respect to the spectral one. However, for one dimensional reaction diffusion equations, the finite order of the spectral discretization is proved to be so high that the convergence turns out to be as fast as one of infinite order.

Recent trends in the numerical solution of retarded functional differential equations

Retarded functional differential equations (RFDEs) form a wide class of evolution equations which share the property that, at any point, the rate of the solution depends on a discrete or distributed set of values attained by the solution itself in the past. Thus the initial problem for RFDEs is an infinite-dimensional problem, taking its theoretical and numerical analysis beyond the classical schemes developed for differential equations with no functional elements. In particular, numerically solving initial problems for RFDEs is a diffcult task that cannot be founded on the mere adaptation of well-known methods for ordinary, partial or integro-differential equations to the presence of retarded arguments. Indeed, efficient codes for their numerical integration need speciffc approaches designed according to the nature of the equation and the behaviour of the solution.By defining the numerical method as a suitable approximation of the solution map of the given equation, we present an original and unifying theory for the convergence and accuracy analysis of the approximate solution. Two particular approaches, both inspired by Runge–Kutta methods, are described. Despite being apparently similar, they are intrinsically different. Indeed, in the presence of speciffc types of functionals on the right-hand side, only one of them can have an explicit character, whereas the other gives rise to an overall procedure which is implicit in any case, even for non-stiff problems.In the panorama of numerical RFDEs, some critical situations have been recently investigated in connection to speciffc classes of equations, such as the accurate location of discontinuity points, the termination and bifurcation of the solutions of neutral equations, with state-dependent delays, the regularization of the equation and the generalization of the solution behind possible termination points, and the treatment of equations stated in the implicit form, which include singularly perturbed problems and delay differential-algebraic equations as well. All these issues are tackled in the last three sections.In this paper we have not considered the important issue of stability, for which we refer the interested reader to the comprehensive book by Bellen and Zennaro (2003).

Stability results for systems described by coupled retarded functional differential equations and functional difference equations

In this work stability results for systems described by coupled Retarded Functional Differential Equations (RFDEs) and Functional Difference Equations (FDEs) are presented. The results are based on the observation that the composite system can be regarded as the feedback interconnection of a subsystem described by RFDEs and a subsystem described by FDEs. Recent small-gain results and Lyapunov-like characterizations of the Weighted Input-to-Output Stability property for systems described by RFDEs and FDEs are employed. It is shown that the stability results provided in this work can be used to study stability for systems described by neutral functional differential equations and systems described by hyperbolic partial differential equations.

Retarded Functional Differential Equations on Manifolds and Applications to Motion Problems for Forced Constrained Systems

Abstract We prove an existence result for T-periodic retarded functional differential equations of the type xʹ(t) = f(t, xt), where f is a T-periodic functional tangent vector field on a smooth manifold. As an application we show that any constrained system acted on by a periodic force, possibly with delay, admits a forced oscillation provided that the constraint is a topologically nontrivial compact manifold and the frictional coefficient is nonzero. We conjecture that the same assertion holds true even in the frictionless case.

Razumikhin-type theorems on stability of stochastic retarded systems

The Razumikhin-type theorems have been well developed and are very effective for the stability of deterministic retarded dynamic systems. Chang (1984) studied the Razumikhin-type stability conditions for stochastic retarded functional differential equations (SRFDEs). But the stability is not in a global sense. Recently, by utilising the Razumikhin-type argument, Mao (1997a) provided some useful criteria for exponential stability of stochastic retarded functional differential equations. However, the important general asymptotic stability has not been considered. The aim of this article is to establish Razumikhin-type theorems on asymptotic stability of stochastic retarded systems, which is a generalisation of the result obtained by Mao. Moreover, generalised exponential stability is considered for better estimate of Lyapunov exponent of time-variant systems. In the conclusion, this article is to show the underlying principle of the Razumikhin-type theorems on stability of stochastic retarded functional differential equations. With this idea, it is not difficult to apply Razumikhin-type techniques to stochastic retarded systems.