An Invariance Principle for Non-smooth Chaotic Dynamical Systems
by
M.A. Aziz Alaoui
Univ. Le Havre, Lab. Applied Math. BP 540 76058 Le Havre Cedex, France
Coauthors: S. Deriviere

The mathematical modeling of physical or biological systems may lead to non-smooth dynamical systems (ODE with discontinuous righthand side). The study of such discontinuous systems uses the theory of Filippov [1] which extends, among other, the concept of solution of differential equations of such a type, and give a seminal contribution for analysing equilibria for discontinuous systems. Thus, the differential equation dx(t)/dt= f(x, t), is replaced by a differential inclusion using a multivalued function F(x, t), such that F(x, t)={ f(x, t) }. Therefore, one have to study the problem : dx(t)/dt belongs to F(x, t). This process is called the convex regularisation, see [1], p.50. Moreover, stability properties of differential systems with discontinuous righthand side using locally lipschitz continuous and regular (in some sense) Lyapunov functions, have been previously done. In this contribution, we present a new result which is an extension of the LaSalle invariance principle for the discontinuous case. It allows us to determine, theoretically, a closed and bounded region in the phase space containing the chaotic attractor for differential equations with discontinuous righthand side, whereas existing theorems concern only the stability of equilibria. The results are obtained in the general context of differential inclusions, an exemple is theoretically and numerically displayed.

Reference: [1] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, 1988.

Date received: March 12, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # canu-44.