Spiralling dynamics near heteroclinic networks

• Construction of a family of vector fields on the 3-sphere with complex dynamics.
• Transverse intersection of invariant manifolds is proved analytically.
• Near a symmetric network, hyperbolic dynamics coexists with heteroclinic tangencies.
• Symmetry-breaking bifurcation from the network creates a spiralling attractor.
• The spiralling attractor is conjugate to a shift over either a finite or an infinite alphabet.

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 S3, 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 phenomena. The vector field is the restriction to S3 of a polynomial vector field in R4. In this article, we also identify global bifurcations that induce chaotic dynamics of different types.

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