IPACS Electronic library

Chaos in multi-valued dynamical systems

Sergej Celikovsky, Zdenek Beran
In our contribution, we address a possible description of chaotic behavior in multi-valued dynamical systems. Non-smooth dynamical systems represent a reasonable part of general multi-valued dynamical systems. They appear widely in a large variety of applications such as mechanics with dry friction, electric circuits with small conductivity, systems with small inertia, economy, biology, control theory, game theory, optimization, etc. There are many different definitions of the concept of “chaos” already in the single-valued dynamical systems. The more the multi-valued dynamical systems are still far from satisfactory realization of that phenomena. The mostly used demonstration of an existence of chaos in case of multi-valued systems uses an appropriate construction of a homeomorphism between one selected solution from the set of them and the bi-directional full shift of symbols, which is, as it is very well known, homeomorphism of the Smale horseshow. In our case, we utilize a result of [1], where is no need to construct a selector on the set of solutions and the definition of chaos therein introduced is much more intuitively descriptive. This concept we demonstrated on some examples of multi-valued dynamical systems and we found the conditions that lead to the chaotic behavior.

[1] 1.Beran Zdeněk : On Characterization of the Solution Set in Case of Generalized Semiflow , Kybernetika vol.45, 5 (2009), p. 701-715 (2009), available at http://www.kybernetika.cz/content/2009/5/701/paper.pdf
File: download
Copyright © 2003—2015 The Laboratory "Control of Complex Systems", IPME RAS