Method of Topological Roughness in Tasks of Research and Control of Synergetic Systems

Abstract

The method of a research of roughness of dynamic systems based on a concept of roughness according to Andronov—Pontryagin and named with "method of topological roughness" is considered. The concepts "roughness" and "bifurcation" of dynamic systems formulated at a dawn of formation of the scientific direction of mathematics — topology, are given by the great French scientist A. Poincare. Also the concept of roughness according to Andronov—Pontryagin is formulated and conditions of accessibility of required roughness of a dynamic system are defined. The definitions of concepts of the maximum roughness and minimum non roughness of dynamic systems entered by the author earlier are given. The corresponding theorems of necessary and sufficient conditions of accessibility of the maximum roughness and the minimum non roughness and also emergence of bifurcations of topological structures of dynamic systems, which were proved in the fundamental works of the author given in the list of references are formulated. At the same time it is claimed that sets of rough and not rough systems make roughnesses of a set, continuous on an indicator. As a roughness indicator in a method the number of conditionality of a matrix of reduction to a diagonal (quasidiagonal) type of a matrix of Jacobi in special points of phase space of a system is used. The method allows to controling roughness of control systems on the basis of the theorem formulated with use of the matrix equation of Sylvester and proved in works of the author which is also provided in this work. The main stages of researches of roughness and bifurcations of systems by means of the considered method are formulated in the form of the corresponding algorithm. In work questions of synergetic systems and chaos (strange attractors) in them, founders of science of synergetics — H. Haken, I. Prigozhin are briefly stated. The method can be used for researches of roughness and bifurcations of dynamic systems and also synergetic systems and chaos of the different physical nature. In works of the author the method is approved on examples of many synergetic systems, such as Lorenz attractors and R ö ssler, Belousov-Zhabotinsky’s systems, Chua, "predator-prey", Henon, Hopf’s bifurcations, etc. In this work of a possibility of a method are illustrated on examples of synergetic system Chua and also a technical system in the form of the nonlinear servomechanism.