Dynamical bifurcation in a system of coupled oscillators with slowly varying parameters

Abstract

This paper deals with a fast-slow system representing n nonlinearly coupled oscillators with slowly varying parameters. We find conditions which guarantee that all omega-limit sets near the slow surface of the system are equilibria and invariant tori of all dimensions not exceeding n, the tori of dimensions less then n being hyperbolic. We show that a typical trajectory demonstrates the following transient process: while its slow component is far from the stationary points of the slow vector field, the fast component exhibits damping oscillations; afterwards, the former component enters and stays in a small neighborhood of some stationary point, and the oscillation amplitude of the latter begins to increase; eventually the trajectory is attracted by an n-dimesional invariant torus and a multi-frequency oscillatory regime is established.

Keywords

• Dynamical bifurcations
• transient processes
• invariant tori
• multi-frequency oscillations
• coupled oscillators
• fast-slow systems