American Institute of Mathematical Sciences

August  2017, 37(8): 4399-4437. doi: 10.3934/dcds.2017189

Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry

 1 Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK 2 Department of Mathematics, Lobachevsky State University of Nizhny Novgorod, 23 Prospekt Gagarina, Nizhny Novgorod 603950, Russia 3 Joseph Meyerhoff Visiting Professor, Weizmann Institute of Science, 234 Herzl Street, Rehovot 7610001, Israel

* Corresponding author: Dongchen Li

Received  February 2017 Revised  February 2017 Published  April 2017

Fund Project: This work was supported by grant RSF 14-41-00044. The authors also acknowledge support by the Royal Society grant IE141468 and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS).

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.

Citation: Dongchen Li, Dmitry V. Turaev. Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4399-4437. doi: 10.3934/dcds.2017189
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The dashed curves represent the two homoclinic loops and the solid vertical lines represent leaves of the foliation $\mathcal{F}_0$. The coincidence condition for system $X$ is that, for any point of $\Gamma^+$ lying in a leaf $l$, there exists one point of $\Gamma^-$ that also lies in $l$
For a four-dimensional system, the intersection points $M^+$ and $M^-$ on a small three-dimensional cross-section $\Pi$ belong to the same leaf of the foliation $\mathcal{F}_1$
As shown in figure (a), we can create an infinite sequence of index-2 point $Q_k^-$ accumulating on $M^-$ while keeping the intersection $W^u(P)\cap W^{ss}(M^-)$ by changing $\mu, \rho$ and $\nu$ together. In figure (b), the intersection $W^u(P)\cap W^s(Q_{k_0}^-)$ is created by changing $\nu$)
The index is determined by the pair ($\lambda_1+\lambda_2, \lambda_1\lambda_2$)
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