Method of topological roughness of dynamic systems: applications to synergetic systems

Abstract

The paper presents a method of dynamic system roughness research, based on Andronov-Pontryagin concept of roughness (method of topological roughness). Andronov-Pontryagin concept of roughness has been formulated. Reachability conditions of dynamic system required roughness are defined. Concept definition for maximum roughness and minimum non-roughness of dynamic systems is given. Theorems on necessary and sufficient conditions of eachability of maximum roughness and minimum non-roughness and occurrence of bifurcations of dynamic system topological structures are formulated. It is claimed that the sets of rough and non-rough systems are continuous in terms of the set roughness. The condition number of the matrix of bringing to the diagonal (quasi-diagonal) view of the Jacobi matrix at special points of the system phase space is used as an indicator of roughness. The method gives the possibility to control the roughness of control systems based on a theorem formulated using Sylvester’s matrix equation. The basic concepts on synergetics and synergetic systems are presented. The method can be used for studies of roughness and bifurcations of dynamic systems, as well as synergetic systems and chaos of various physical nature. The method is tested on the examples of many synergetic systems: Lorenz and Rössler, Belousov-Zhabotinsky, Chua, “predator-prey”, Henon, and Hopf bifurcation. The main provisions of the topological roughness method are given. The possibilities of the method are illustrated by examples of Belousov-Zhabotinsky and Chua synergetic systems.