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arXiv · 1304.5283

Spiralling dynamics near heteroclinic networks

Alexandre A. P. Rodrigues·Isabel S. Labouriau
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Abstract

There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. We construct explicitly a {two parameter family of vector fields} on the three-dimensional sphere $\EU^3$, whose flow has a spiralling attractor containing the following: two hyperbolic equilibria, heteroclinic trajectories connecting them {transversely} and a non-trivial hyperbolic, invariant and transitive set. The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The vector field is the restriction to $\EU^3$ of a polynomial vector field in $\RR^4$. In this article, we also identify global bifurcations that induce chaotic dynamics of different types.

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Alexandre A. P. Rodrigues, Isabel S. Labouriau. 2013-04-18. Spiralling dynamics near heteroclinic networks. https://doi.org/10.1016/j.physd.2013.10.012

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