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

  • * Corresponding author: Dongchen Li

    * Corresponding author: Dongchen Li 
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).
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  • 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.

    Mathematics Subject Classification: Primary: 37G20, 37G25, 37G35.


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  • Figure 1.  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 $l$

    Figure 2.  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 $\mathcal{F}_1$

    Figure 3.  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 $\nu$)

    Figure 4.  The index is determined by the pair ($\lambda_1+\lambda_2, \lambda_1\lambda_2$)

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