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Structurally stable heteroclinic cycles

Published online by Cambridge University Press:  24 October 2008

John Guckenheimer  and
Philip Holmes
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.
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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 G ⊂ O(3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 189 - 192
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
[2]Aubry, N., Holmes, P. J., Lumley, J. L. and Stone, E.. The dynamics of coherent structures in the wall region of a turbulent boundary layer. (Submitted for publication.)Google Scholar
[3]Busse, F. H. and Heikes, K. E.. Convection in a rotating layer: a simple case of turbulence. Science 208 (1980), 173–175.CrossRefGoogle Scholar
[4]Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, 1983).CrossRefGoogle Scholar
[5]Nicolaenko, B. (personal communication).Google Scholar