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 , the group orbit of (under rotations) gives a whole (invariant) manifold of equilibria. Under generic conditions we have that, after perturbing our equation by a (small) -equivariant perturbation, persists as an invariant manifold . However, the flow on is in general no longer trivial. Indeed, we find slow dynamics on and, in the case (the tetrahedral subgroup of ), 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