Existence of Heterodimensional Cycles near Shilnikov Loops

Dongchen Li, Dmitry V. Turaev

Published 2015-12-03Version 1

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