TOPOLOGICAL ROUGHNESS OF SYNERGETIC SYSTEMS

Abstract

The method of investigation of roughness of dynamic systems, based on the concept of roughness on Andronov-Pontryagin and called "method of topological roughness," is considered. The definitions of the concepts of maximum roughness and minimum non-roughness of dynamic systems are given. The corresponding theorems on the necessary and sufficient conditions of reachability of maximum roughness and minimum non-roughness, as well as the occurrence of bifurcations of topological structures of dynamic systems, which have been proved in the author 's works given in the references, have been formulated. The method allows to control the roughness of control systems based on the theorem formulated using the matrix Sylvester equation. The method can be used for studies of roughness and bifurcations of dynamic systems, as well as synergetic systems and chaos of different physical nature. In the author 's works, the method is tested on the examples of many synergetic systems, such as Lorenz and Rossler attractions, Belousov-Zhabotinsky, Chua, predator-prey, Henon, Hopf bifurcation, Kaldor and Sсhumpeter economic systems etc. The applications of the method are illustrated by the example of the Chua synergetic system.