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Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in \({\mathbb R}^4\)

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Abstract

Robust heteroclinic cycles in equivariant dynamical systems in \({\mathbb R}^4\) have been a subject of intense scientific investigation because, unlike heteroclinic cycles in \({\mathbb R}^3\), they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyze the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in \({\mathbb R}^4\).

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Notes

  1. The isotypic decomposition of the representation of a group is the (unique) decomposition in a sum of equivalence classes of irreducible representations.

  2. Since \(\ddot{r}=\dot{r}(\alpha +2\beta r\cos 3\theta )+3\beta ^2r^3\sin ^23\theta \), an initially positive \(\dot{r}\) remains positive for all \(t>0\). Therefore, r is an increasing function of t for small initial \(r_0\) and, hence, we can consider \(\theta \) as a function of r.

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Correspondence to Olga Podvigina.

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Communicated by Eusebius Doedel.

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Podvigina, O., Chossat, P. Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in \({\mathbb R}^4\) . J Nonlinear Sci 27, 343–375 (2017). https://doi.org/10.1007/s00332-016-9335-4

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  • DOI: https://doi.org/10.1007/s00332-016-9335-4

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