Hildebrando M. Rodrigues

On the invariance principle: generalizations and applications to synchronization

In many engineering and physics problems it is very hard to find a Lyapunov function satisfying the classical version of the LaSalles invariance principle. In this work, an extension of the invariance principle, which includes cases where the derivative of the Lyapunov function along the solutions is positive on a bounded set, is given. As a consequence, a larger class of problems may now be considered. The results are used to obtain estimates of attractors which are independent of coupling parameters. They are also applied to study the synchronization of coupled systems, such as coupled power systems and coupled Lorenz systems. Estimates on the coupling term are obtained in order to accomplish the synchronization.

Abstract methods for synchronization and applications

In this work we present some mathematical methods to obtain uniform estimates for attractors and to study the synchronization of two similar dynamical systems. We give sufficient conditions to control the coupling devices in order to accomplish the synchronization, even when each of the systems moves chaotically. Some examples that include systems of coupled lasers and coupled Lorenz equations are discussed.

Properties of Bounded Solutions of Linear and Nonlinear Evolution Equations: Homoclinics of a Beam Equation*

Abstract The objective of this work is to discuss the existence, bifurcation, and regularity, with respect to time and parameters, of bounded solutions of infinite dimensional equations. The authors present an application of their results to the study of homoclinic solutions of a nonlinear forced beam equation. They use the approach of alternative method and semigroup theory.

Applications of robust synchronization to communication systems

In this work, using chaotic systems, we study the role of synchronization on codification and decodification of messages. We first present a general result that is useful to prove uniform dessipativeness for nonautonomous systems of ordinary differential equations. Then some theorems are established to give sufficient conditions to obtain synchronization of coupled systems. The above results are applied to some specfic coupled systems, namely, coupled Lorenz systems, coupled Duffings equations, coupled Chuas systems, etc., showing how to code and decode message using chaotic systems. One of our main results is to obtain the robustness of the synchronization with respect to parameter variation.

On the relationship between exponential dichotomies and the Fredholm Alternative

The above Fredholm Alternative is very important in studying homoclinic and heteroclinic solutions of nonlinear differential equations, using the Liapunov-Schmidt Method. This was pointed out by Chow, Hale, and Mallet-Paret in [3 3 and since then many results appeared even in the infinite dimensional case. The last ones can be found in Blasquez [l], Hale and Lin [S], Lin [7], Rodrigues and Silveira [lo], and Silveira [ 111. 2.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

On harmonic and subharmonic solutions of nonlinear second order equations: symmetry and bifurcation

Consider the equation , where p, y are small parameters, f is an even continuous π/m-odd- harmonic function (i.e., f(t + π/m)= -f(t), for every t in R)m≧2 and g is an odd function of u. Under certain conditions on f and g it is proved that the small 2π-periodic solutions of the above equation maintain some symmetry properties of the forcing f(t), when μ≠0. Other interesting results describe the changes of the number of such solutions, as p and μ vary in a small neighborhood of the origin. As another contribution of this paper, it was proved that a central assumption which was required in the main results, is generic. The main tool used in this work is the Liapunov-Schmidt Method.

Relative Asymptotic Equivalence with Weight tμ, between Two Systems of Ordinary Differential Equations

Publisher Summary This chapter describes the relative asymptotic equivalence with weight t μ between two systems of ordinary differential equations. The chapter presents a comparison of the solutions of the linear system y = A ( t ) y with the solutions of the perturbed linear system x = A ( t ) x + f ( t, x ), where x ∈ X , with X = R n or X = C n , A ( t ) is an n ɛn matrix continuous on J = [ t 0 , ∞ ), t 0 ≥ 0 , and f ( t, x ) is an n-dimensional column vector function continuous on J x X.

Um Princípio de Invariância Uniforme: robustez com relação à variação de parâmetros

O objetivo deste trabalho e a obtencao de estimativas uniformes, com relacao aos parâmetros, do atrator e da area de atracao de um sistema dinâmico e a aplicacao destes resultados a analise da robustez da sincronizacao de dois subsistemas acoplados. Estas estimativas sao obtidas atraves de uma versao uniforme do Principio de Invariância de LaSalle o qual e proposto e demonstrado neste trabalho.