Symmetric chaos in a local codimension two bifurcation with the symmetry group of a square

SIAM J Appl. Dyn. Sys 4 (2005) 32-52

S. Abreu, P. Aston and I. Melbourne

Abstract

We study a codimension two steady-state/steady-state mode interaction with D₄ symmetry, where the centre manifold is three-dimensional. Primary branches of equilibria undergo secondary Hopf bifurcation to periodic solutions which undergo tertiary bifurcations leading to chaotic dynamics. This is not an exponentially small effect, and the chaos obtained in simulations using DsTool is large-scale, in contrast to the `weak' chaos associated with Shilnikov theory.

Moreover, there is an abundance of symmetric chaotic attractors and symmetry-increasing bifurcations. Numerical investigations demonstrate that the symmetric chaos is part of the local codimension two bifurcation. The two-dimensional parameter space is mapped out in detail for a specific choice of Taylor coefficients for the centre manifold vector field. We use AUTO to compute the transitions involving periodic solutions, Lyapunov exponents to determine the chaotic region, and symmetry detectives to determine the symmetries of the various attractors.