Multistability and memory effects in dynamical system with cosymmetric potential

Abstract

The purpose of present study is the analysis of strong multistability in a dynamical system with cosymmetry. We study the dynamics and realization of steady-states in a mechanical system with two degrees of freedom. The minimum potential energy of the system is achieved on a curve in the form of an ellipse, which gives rise to a continuum family of equilibria and strong multistability. This problem belongs to the class of dynamical systems with cosymmetry. Methods. To analyze the system, we used methods of computational qualitative analysis of dynamical systems and cosymmetry theory. Results. The behavior of the system is studied for different values of the initial potential energy, semi-axes of the ellipse and the friction coefficient. In the conservative cosymmetric case, the existence of chaotic regions in the phase space is established. In the presence of friction, a complex dependence of the realization of equilibria on the family on the initial data was numerically found, which is due to the memory effect of conservative chaos. The results of the analysis of the system in case of violation of cosymmetry are presented and the memory effects of the destroyed family of equilibria are demonstrated. Conclusion. With strong multistability, the effects of memory about the properties of the system under a small violation of them have a significant effect on the dynamics. Despite the determinism in the presence of friction (all trajectories tend to equilibria), there is a strong dependence of the implementation of equilibria on the initial data, which is characteristic of chaos. With a small violation of cosymmetry, the system also demonstrates the memory of the disappeared continual family of equilibria: from all the initial data, the trajectories first tend to the neighborhood of the family, and then slowly drift along it to one of the remaining equilibria.

Keywords

• nonlinearity
• nonlinear systems
• differential equations
• multistability
• cosymmetry
• chaos
• selection