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An Isaac Newton Institute Workshop

Theory and Applications of Coupled Cell Networks

Symmetric Chaos in a Local Codimension Two Bifurcation

Authors: Stella Abreu (Univ. Portucalense), Plilip Aston (Univ. Surrey), Ian Melbourne (Univ. Surrey)


We study a codimension two steady-state/steady-state mode interaction with D4 symmetry, where the center manifold is three-dimensional. Primary branches of equilibria undergo secondary Hopf bifurcations to periodic solution which undergo further 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. The local bifurcation studied by us is the simplest (in terms of dimension of the center manifold and codimension of the bifurcation) in which such fenomena have been identified. Numerical investigations demonstrate that the symmetric chaos is part of the local codimension two bifurcation. The two dimensional parameter space is maped out in detail for a specific choice of Taylor coefficients for the center 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.

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