Research of the stability of some hereditary dynamic systems

Abstract

In the training course of the theory of differential equations, there exists a section on the investigation of the stability of systems of differential equations. If the system of differential equations consists of differential equations of integer order, then the stability theory of Lyapunov is usually used to research the stability of their rest points. However, in the case when the system of differential equations consists of differential equations of non-integer order, then it is necessary to use other methods of investigating the stability of such systems. Therefore, this article is devoted to the method of investigating the rest points of systems of differential equations of fractional order. In this paper we will investigate the stability of the rest points of the hereditary dynamical systems by the example of some fractal oscillators. Moreover, we will consider two types of hereditary dynamical systems: commensurable and incommensurate, for which the corresponding stability theorems for rest points are valid. Next, examples of applying these stability theorems to a fractal linear oscillator, the Duffing fractal oscillator, will be considered. The results of the research of the stability of the rest points of the hereditary dynamical systems were confirmed by constructing phase trajectories for the fractal oscillators under consideration. This article can be useful in the study of a fairly new section in the theory of differential equations-fractional calculus.