Periodic perturbations of codimension-two bifurcations with a double zero eigenvalue in dynamical systems with symmetry

TL;DR

This paper analyzes bifurcation phenomena in symmetric dynamical systems with double zero eigenvalues under small periodic perturbations, revealing complex behaviors like chaos and providing methods applicable to high-dimensional systems.

Contribution

It introduces a method to analyze bifurcations with symmetry and double eigenvalues under periodic perturbations, extending to infinite-dimensional systems.

Findings

Existence of transverse homoclinic and heteroclinic orbits

Chaotic dynamics in wide parameter regions

Validation through numerical simulations with AUTO

Abstract

We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed system to a simpler one which is a periodic perturbation of the normal form for codimension-two bifurcations with a double zero eigenvalue and symmetry, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring in the system. In particular, we show that there exist transverse homoclinic or heteroclinic orbits, which yield chaotic dynamics, in wide parameter regions. These results can be applied to three or higher-dimensional systems and even to infinite-dimensional systems with the assistance of center manifold reduction and the invariant manifold theory. We illustrate our theory for a pendulum subjected to position and velocity feedback control when the desired position is periodic in time. We also give numerical computations by the computer tool AUTO to demonstrate the theoretical results.