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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Heteroclinic cycles for reaction diffusion systems by forced symmetry-breaking


Authors: Stanislaus Maier-Paape and Reiner Lauterbach
Journal: Trans. Amer. Math. Soc. 352 (2000), 2937-2991
MSC (2000): Primary 37G40; Secondary 35B32, 34C14, 58K70
DOI: https://doi.org/10.1090/S0002-9947-00-02311-4
Published electronically: March 27, 2000
MathSciNet review: 1604002
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Abstract: Recently it has been observed, that perturbations of symmetric ODE's can lead to highly nontrivial dynamics. In this paper we want to establish a similar result for certain nonlinear partial differential systems. Our results are applied to equations which are motivated from chemical reactions. In fact we show that the theory applies to the Brusselator on a sphere. To be more precise, we consider solutions of a semi-linear parabolic equation on the 2-sphere. When this equation has an axisymmetric equilibrium $u_\alpha$, the group orbit of $u_\alpha$ (under rotations) gives a whole (invariant) manifold $M$ of equilibria. Under generic conditions we have that, after perturbing our equation by a (small) $L\subset {{{\bf O}(3)}}$-equivariant perturbation, $M$ persists as an invariant manifold $\widetilde M$. However, the flow on $\widetilde M$ is in general no longer trivial. Indeed, we find slow dynamics on $\widetilde M$ and, in the case $L=\mathbb{T} $ (the tetrahedral subgroup of ${{{\bf O}(3)}}$), we observe heteroclinic cycles. In the application to chemical systems we would expect intermittent behaviour. However, for the Brusselator equations this phenomenon is not stable. In order to see it in a physically relevant situation we need to introduce further terms to get a higher codimension bifurcation.


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Additional Information

Stanislaus Maier-Paape
Affiliation: Institut für Mathematik, Universität Augsburg, Universitätsstraße 14, D-86135 Augsburg, Germany
Email: maier@math.uni-augsburg.de

Reiner Lauterbach
Affiliation: Institute for Applied Mathematics, University of Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany
Email: lauterbach@math.uni-hamburg.de

DOI: https://doi.org/10.1090/S0002-9947-00-02311-4
Received by editor(s): November 22, 1995
Received by editor(s) in revised form: October 8, 1997
Published electronically: March 27, 2000
Article copyright: © Copyright 2000 American Mathematical Society