Applications of the method of topological roughness to the study of synergetic systems of geodynamics (PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE KYRGYZ REPUBLIC, 2024)
A method for studying the roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin and called the «method of topological roughness» is considered. The definitions of the concepts of maximum roughness and minimum roughness of dynamic systems are given, the corresponding theorems are formulated on the necessary and sufficient conditions for their attainability, as well as the occurrence of bifurcations of topological structures of dynamic systems, which were proved in the author’s works. The method allows you to control the roughness of systems based on the theorem formulated using the Sylvester matrix equation at singular points of the phase space. The main stages of studying the roughness and bifurcations of systems using the considered method are formulated in the form of an appropriate algorithm. The method can be used to study the roughness and bifurcations of dynamic systems, as well as synergetic systems and chaos of various physical nature. In the works of the author, the method was tested on the examples of many synergetic systems, such as attractors of Lorenz and Rössler, Belousov-Zhabotinsky, Chua, «predator-prey», Henon, Hopf bifurcations, etc. In this paper, the method is used to study geodynamic synergetic processes in the area preparing strong earthquakes.
Keywords: topological roughness of dynamical systems; synergetic systems and chaos; singular points of phase space; Sylvester matrix equation; condition number of matrices