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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 103, Issue 1
  • January 1988, pp. 189-192

Structurally stable heteroclinic cycles

  • John Guckenheimer (a1) and Philip Holmes (a1)
  • DOI: http://dx.doi.org/10.1017/S0305004100064732
  • Published online: 24 October 2008
Abstract

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.

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[3]F. H. Busse and K. E. Heikes . Convection in a rotating layer: a simple case of turbulence. Science 208 (1980), 173175.

[4]J. Guckenheimer and P. Holmes . Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, 1983).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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