Dynamics of a Mechanical System with Curve of Equilibria: Cosymmetry and Multistability

Abstract

We analyze the effects of extreme multistability in a mechanical system which describes the movement of an idealized ball over a surface like a Mexican hat. The mathematical model is given by a system of autonomous ordinary differential equations with parameters. In particular cases of rotational symmetry and cosymmetry, the system has a curve of asymptotically stable equilibria. The symmetry gives a circle of equilibria with identical stability spectra, whereas the cosymmetry produces an ellipse of equilibria with nonidentical properties. The destruction of both symmetry and cosymmetry leads to a finite number of equilibria (multistability). We study the dynamics for conservative (without dissipation) and dissipative (linear damping) cases using analytical methods and computer simulation. We found interesting effects caused by extreme multistability: nontrivial selection of equilibria of the family, high sensitivity to initial data because of memory about conservative chaos, and essential difference in dynamics in rotational symmetry and cosymmetry cases.