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Existence of Heterodimensional Cycles near Shilnikov Loops in Systems with a $\mathbb{Z}_2$ Symmetry

Authors
  • Li, Dongchen
  • Turaev, Dmitry V.
Type
Preprint
Publication Date
Oct 22, 2016
Submission Date
Dec 03, 2015
Identifiers
DOI: 10.3934/dcds.2017189
Source
arXiv
License
Yellow
External links

Abstract

We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.

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