Periodic Differential Systems Research Articles

Time symmetries and in-period transformations

A theorem has been proved which gives us an opportunity to investigate the properties of in-period transformations of periodic differential systems.

Reduction of periodic difference systems to linear or autonomous ones

We extend Floquet theory for reducing nonlinear periodic difference systems to autonomous ones (actually linear) by using normal form theory.

The embedding flows of [formula omitted] hyperbolic diffeomorphisms

In [Weigu Li, J. Llibre, Xiang Zhang, Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows, Amer. J. Math. 124 (2002) 107–127] we proved that for a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) in (Rn,0), if A has a real logarithm with its eigenvalues weakly nonresonant, then F(x) can be embedded in a C∞ autonomous differential system. Its proof was very complicated, which involved the existence of embedding periodic vector field of F(x) and the extension of the Floquet's theory to nonlinear C∞ periodic differential systems. In this paper we shall provide a simple and direct proof to this last result.Next we shall show that the weakly nonresonant condition in the last result on the real logarithm of A is necessary for some C∞ diffeomorphisms F(x)=Ax+f(x) to have C∞ embedding flows.Finally we shall prove that a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) with f(x)=O(|x|2) in (R2,0) has a C∞ embedding flow if and only if either A has no negative eigenvalues or A has two equal negative eigenvalues and it can be diagonalizable.

Recent results on the stability of time dependent sets and their application to bifurcation problems

In the first part of the paper we give a short review of our recent results concerning the relationship between conditional and unconditional stability properties of time dependent sets, under smooth differential systems in \mathbf R^n . More precisely, let M be an ‘‘ s -compact’’ invariant set in \mathbf R \times \mathbf R^n and let Φ be a smooth invariant set in \mathbf R \times \mathbf R^n containing M . It is assumed that M is uniformly asymptotically stable with respect to the perturbations lying on Φ . The unconditional stability properties of M depend on the stability properties of Φ ‘‘near M ’’. This dependence has been analyzed in general, and, in the periodic case, complete characterizations are obtained. In the second part, the above results have been applied to bifurcation problems for periodic differential systems. Some our previous statements on the matter are revisited and enriched.

Polynomial and linearized normal forms for almost periodic difference systems

In this paper, we prove the analytic conjugacy theorems of Poincaré type for almost periodic difference systems based on the Lyapunov exponents of corresponding reduced systems.

On the qualitative behavior of periodic solutions of differential systems

This article deals with the reflective function of differential systems. The obtained results are applied to studying the existence and stability of the periodic solutions of some linear and nonlinear periodic differential systems.

Normal forms for almost periodic differential systems

AbstractIn this paper we prove smooth conjugate theorems of Sternberg type for almost periodic differential systems, based on the Lyapunov exponents of the corresponding reduced systems.

Positive periodic solutions of nonautonomous functional differential systems

Considered is the periodic functional differential system with a parameter, x ′ ( t ) = A ( t , x ( t ) ) x ( t ) + λ f ( t , x t ) . Using the eigenvalue problems of completely continuous operators, we establish some criteria on the existence of positive periodic solutions. Moreover, we apply the results to a couple of population models and obtain sufficient conditions for the existence of positive periodic solutions, which are compared with existing ones.

Periodic solutions of nonlinear periodic differential systems with a small parameter

We deal with nonlinear periodic differential systems depending on a small parameter. Theunperturbed system has an invariant manifold of periodic solutions. We provide sufficientconditions in order that some of the periodic orbits of this invariant manifold persist after theperturbation. These conditions are not difficult to check, as we show in some applications. Thekey tool for proving the main result is the Lyapunov--Schmidt reduction method applied to thePoincaré--Andronov mapping.

Differential systems, the mapping over period for which is represented by a product of three exponential matrixes

For some periodic differential systems the sufficient conditions of a representation of the mapping over period as a product of a three exponential matrix of the special aspect are obtained. Obtained results are used for research of problems of the existence and stability of periodic solutions of nonlinear systems and the asymptotic orbital stability of cycles of autonomous differential systems.

Almost periodic linear differential equations with non-separated solutions

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov–Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov–Levitan together with the theory of characters in topological groups.

On the singular and higher characteristic exponents of an almost periodic linear differential system with an affine dependence on a parameter

On the singular and higher characteristic exponents of an almost periodic linear differential system with an affine dependence on a parameter

THE COMPUTATION OF LYAPUNOV EXPONENTS FOR PERIODIC TRAJECTORIES

We present a new method for the numerical computation of Lyapunov exponents of periodic trajectories which is crucial in the investigation of dynamics. The computational time of this method is merely a period of the periodic trajectory considered, when Lyapunov exponents can be approximated as precise as one wants. Our method stems from a combination between Floquet theory on periodic linear differential systems and Lie group methods in structure preserving algorithms on manifolds. The Lyapunov exponents of a periodic trajectory of the Lorenz system and a periodic solitary wave of the nonlinear Schrödinger equation with periodic coefficients are investigated by using the method.

On Partially Irregular Almost Periodic Solutions of Weakly Nonlinear Ordinary Differential Systems

For weakly nonlinear almost periodic ordinary differential systems, we obtain conditions for the existence of partially irregular almost periodic solutions and propose algorithms for their construction.

On the stability in periodic and almost periodic difference systems

In this paper we consider the system of difference equations x n + 1 = f n ( x n ) , f n ( 0 ) = 0 , n = 0 , 1 , 2 , … . This system has the trivial (zero) solution x n = 0 . Sufficient conditions of its asymptotic stability are obtained in the cases when functions f n ( x ) are periodic and almost periodic in n.

Small parameter perturbations of nonlinear periodic systems**Supported by the National Research Project MIUR: ‘Feedback control and optimal control’, by RFBR grants 02-01-00189, 02-01-00307, 03-01-06308 and USCRDF—RF Ministry of Education grant VZ-010.

In this paper we consider a class of nonlinear periodic differential systems perturbed by two nonlinear periodic terms with multiplicative different powers of a small parameter ε > 0. For such a class of systems we provide conditions that guarantee the existence of periodic solutions of given period T > 0. These conditions are expressed in terms of the behaviour on the boundary of an open bounded set U of of the solutions of suitably defined linearized systems. The approach is based on the classical theory of the topological degree for compact vector fields. An application to the existence of periodic solutions to the van der Pol equation is also presented.

Local first integrals of differential systems and diffeomorphisms

In this paper using theory of linear operators and normal forms we generalize a result of Poincare [11] about the non-existence of local first integrals for systems of differential equations in a neighbourhood of a singular point. As an application of the generalized result, and under more weak conditions we obtain a result of Furta [8] about local first integrals of semiquasi-homogeneous systems. Moreover, for diffeomorphisms and periodic differential systems we give definitions of their first integrals, and generalize the previous results about systems of differential equations to diffeomorphisms in a neighbourhood of a fixed point and to periodic differential systems in a neighbourhood of a constant solution.

Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows

This paper has two parts. In the first one we generalize the Floquet theory to nonlinear periodic differential systems. In the second part we apply the first one to obtain a sufficient condition in order that a germ of diffeomorphism can be embedded in a differential flow.

Applications of a Zp Index Theory to Periodic Solutions for a Class of Functional Differential Equations

By means of a new geometrical index with Zp group actions, multiplicity results for a certain class of nonautonomous time periodic functional differential systems are obtained.

Exponential Dichotomies, Almost Periodic Structurally Stable Differential Systems, and an Example

In this paper, we introduce concepts of almost periodic topological (in short, APT) equivalence and almost periodic structural (in short, APS) stability of differential systems, and a necessary and sufficient condition for APS stability of differential systems. Furthermore, we discuss the inheritability of the APS stability and density of APS stable differential systems in a certain space of uniformly almost periodic differential systems.