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Symmetric Chaos in a Local Codimension Two Bifurcation with the Symmetry Group of a Square

Abreu, Stella, Aston, Philip J. and Melbourne, Ian (2005) Symmetric Chaos in a Local Codimension Two Bifurcation with the Symmetry Group of a Square SIAM Journal on Applied Dynamical Systems, 4. pp. 32-52.


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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 solutions 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 in this paper is the simplest (in terms of dimension of the center manifold and codimension of the bifurcation) in which such phenomena have been identified. 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 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.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Abreu, Stella
Aston, Philip
Date : 14 January 2005
DOI : 10.1137/040603310
Uncontrolled Keywords : symmetric chaos, symmetry-increasing bifurcations, local bifurcation theory, mode interaction
Additional Information : Published in the SIAM journal of Applied and Dynamical Systems, Vol 4 (1) pp.32-52. © 2005, Society for Industrial and Applied Mathematics.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:40
Last Modified : 29 Aug 2019 12:58

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