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Structurally stable heteroclinic cycles

Published online by Cambridge University Press:  24 October 2008

John Guckenheimer
Affiliation:
Departments of Mathematics and Theoretical and Applied Mechanics and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.
Philip Holmes
Affiliation:
Departments of Mathematics and Theoretical and Applied Mechanics and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.

Extract

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ3 equivariant with respect to a particular finite subgroup GO(3) such that each XU has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Armbruster, D., Guckenheimer, J. and Holmes, P.. Heteroclinic cycles and modulated travelling waves in systems with O (2) symmetry. Physica D, to appear.Google Scholar
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