On the Bifurcations of a 3D Symmetric Dynamical System

Abstract

The paper studies the bifurcations that occur in the T-system, a 3D dynamical system symmetric in respect to the Oz axis. Results concerning some local bifurcations (pitchfork and Hopf bifurcation) are presented and our attention is focused on a special bifurcation, when the system has infinitely many equilibrium points. It is shown that, at the bifurcation limit, the phase space is foliated by infinitely many invariant surfaces, each of them containing two equilibrium points (an attractor and a saddle). For values of the bifurcation parameter close to the bifurcation limit, the study of the system’s dynamics is done according to the singular perturbation theory. The dynamics is characterized by mixed mode oscillations (also called fast-slow oscillations or oscillations-relaxations) and a finite number of equilibrium points. The specific features of the bifurcation are highlighted and explained. The influence of the pitchfork and Hopf bifurcations on the fast-slow dynamics is also pointed out.