Examples of forced symmetry-breaking to homoclinic cycles in three-dimensional Euclidean-invariant systems

Parker, M. J.; Stewart, I. N.; Gomes, M. G. M.

http://hdl.handle.net/10400.7/45

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Authors

Parker, M. J.

Stewart, I. N.

Gomes, M. G. M.

Abstract(s)

We study perturbations of cubic planforms, proving there exists perturbations with homoclinic cycles between persistent steady states. Our results do not depend on the representation of the symmetry group of the lattice, and are thus quite general. . The problem is studied using group theory rather than direct methods. We use the abstract action of the symmetry group of the perturbation on the group orbit to determine the existence of zero- and one-dimensional flow-invariant subspaces. The residual symmetry of the perturbation constrains the flows on these subspaces and, in certain cases, homoclinic cycles are guaranteed to exist. Cubic planforms are physically interesting due to their relevance to certain physical systems. Applications to reaction-diffusion systems, nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction are discussed

Keywords

forced symmetry-breaking homoclinic Euclidean symmetry reaction-diffusion-systems oxygen oscillating reaction

URI

http://hdl.handle.net/10400.7/45

Citation

Parker, M.J., Stewart, I., Gomes, M.G.M.(2008)."Examples of forced symmetry-breaking to homoclinic cycles in three-dimensional Euclidean-invariant systems". International Journal of Bifurcation and Chaos.18(1): 83-107