Autonomous Retarded Functional Differential Equations Research Articles

On polynomial forms of nonlinear functional differential equations

<p style='text-indent:20px;'>In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.</p>

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.

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.

Symbolic computation for center manifolds and normal forms of Bogdanov bifurcation in retarded functional differential equations

In this paper, we present explicit formulas for computing the coefficients of a center manifold for the Bogdanov singularity in autonomous retarded functional differential equations with parameters. As a consequence, normal forms associated with the flow on a center manifold up to an arbitrary order are derived. The explicit formulas have been implemented using the computer algebra system Maple. The paper ends with some examples given in order to show the applicability of the methodology and the convenience of the computer software.

Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity

In this paper, we present explicit formulas for computing the coefficients of a center manifolds for the Fold-Hopf singularity in autonomous retarded functional differential equations. As consequence, normal forms associated with the flow on a center manifold up to an arbitrary order are derived. The explicit formulas have been implemented using the computer algebra system Maple. We apply our results to a delayed system in order to show the applicability of the methodology.

A Maple program for computing a terms of a center manifold, and element of bifurcations for a class of retarded functional differential equations with Hopf singularity

In this paper, we present explicit formulas for computing the coefficients of a center manifold for the Hopf singularity in autonomous retarded functional differential equations. As consequence, an algorithm for computing Lyapunov constant and normal forms up to an arbitrary order is derived. The explicit formulas have been implemented using the computer algebra system Maple. The paper ends with some examples in order to show the applicability of the methodology and the convenience of the computer software.

Une généralisation du théorème de Cartwright

In this work we first explain a reduced method developed by R.A. Smith, then we apply it to extend a theorem due to M. Cartwright to a class of autonomous retarded functional differential equations. We show, for these equations, that the almost-periodic solutions, when they exist, are quasi-periodic.

Normal forms and Hopf bifurcation for partial differential equations with delays

The paper addresses the computation of normal forms for some Partial Functional Differential Equations (PFDEs) near equilibria. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations and on the existence of center (or other invariant) manifolds. As an illustration of this procedure, two examples of PFDEs where a Hopf singularity occurs on the center manifold are considered.

Normal forms for periodic retarded functional differential equations

This paper addresses the computation of normal forms for periodic retarded functional differential equations (FDEs) with autonomous linear part. The analysis is based on the theory previously developed for autonomous retarded FDEs. Adequate nonresonance conditions are derived. As an illustration, the Bogdanov–Takens and the Hopf singularities are considered.

A One Parameter Family of Spline-Type Schemes for Approximation of Delay Systems

This paper presents a one parameter family of approximation schemes for systems of linear autonomous retarded functional differential equations. The schemes are based on transformation of the given system into an abstract Cauchy problem on the state space Rn × L2 and on approximation of the corresponding solution semigroup on sequences of finite dimensional subspaces of the state space. The parameter ranges over a certain real interval, whose extreme points are determined by the values generating an orthogonal basis consisting of piecewise linear functions respectively a basis of linear splines for the finite dimensional subspaces. This construction of the one parameter family is used to investigate the eigenvalue behaviour of linear spline approximating systems and to compare it with that of other approximating systems of the family. Moreover, a new approach for proving consistency is presented, showing and utilizing a close relationship between approximating schemes for delay systems and ODE-methods.

On periodic solutions to autonomous retarded functional differential equations

Under the assumption that Ca = C([−r, 0], Sn−1(a)) is positively invariant for a > 0, two necessary and sufficient conditions are obtained for an autonomous retarded functional differential equation to have a non-trivial periodic solution in Ca. Moreover, a feasible sufficient condition is given, which is better for n = 2 than that given by Dos Reis and Baroni.

On the existence of periodic solutions for autonomous retarded functional differential equations on R2

SynopsisLet Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

Existence of periodic orbits of autonomous retarded functional differential equations

For proving the existence of periodic orbits of autonomous ordinary differential equations, three different methods are available, namely, the Hopf bifurcation theorem, the torus principle and the Poincaré–Bendixson theorem. Until recently, the Poincaré–Bendixson theorem was applicable only to equations of the second order. However, it was extended in (13, 14) to certain equations of higher order by means of a plane projection technique. In §§ 2, 3 of the present paper this technique is adapted to prove analogues of the Poincaré–Bendixson theorem for a class of autonomous retarded functional differential equations. For such equations the orbits lie in a Banach space studied by Hale ((7), p. 43). The main hypothesis of this theorem is that the equation has a bounded semi-orbit whose ω-limit set contains no critical points. The problem of finding such a semi-orbit is solved in § 4 for a class of dissipative equations. The practical application of these results requires the construction of certain functionals which are similar to the Liapunov functionals of stability theory. In § 5, these functionals are constructed for a large class of retarded feedback control equations and explicit conditions for the existence of periodic orbits are deduced. Some practical details are illustrated in § 6 by applying the theory to certain scalar delay-differential equations.

Completeness and F-completeness of eigenfunctions associated with retarded functional differential equations

The paper gives general necessary and sufficient conditions for completeness of generalized eigenfunctions associated with systems of linear autonomous retarded functional differential equations (FDE), in the Hilbert space R n × L 2([− h, 0], R n ), and also in the space C([− h, 0], R n ). The generalized eigenfunctions are elements of the nullspaces of ( Iλ − A) m , where A is the infinitesimal generator of a C 0-semigroup of bounded linear operators on R n × L 2([− h, 0], R n ) (resp. C([− h, 0], R n )) corresponding to the FDE in question. In addition to the usual notion of completeness, a new concept of F-completeness is introduced and its significance is explained. In particular, it is shown that the F-completeness is related to the absence of solutions of the transposed equation that vanish in finite time. The results are obtained entirely via the C 0-semigroup theory, which results in simplicity of the proofs. As a by-product, some new results on the adjoint semigroup are obtained. The main results are expressed in an operator form. These are translated into conditions expressed in terms of the original system matrices. For systems with one delay, the F-completeness criterion is translated into a verifiable matrix type criterion, in which the concepts of maximal controllability and invariant subspaces of two matrices play a prominent role.

Stability of attractivity regions for autonomous functional differential equations

We consider attractivity regions of the zero solutions of autonomous retarded functional differential equations (FDEs). It is shown that the attractivity region remains unchanged if the FDE undergoes a small perturbation in certain classes of FDEs. Especially we obtain the openness of the set of parameters α∈(0,π/2) such that the attractivity region of the equation\(\dot x(t) = - \alpha x(t - 1)[1 + x(t)]\) is given by the set of continuous functions ϕ:[−1,0] → R with ϕ(0)>−1.