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

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.

43 pages, 4 figures

References in corpus (2)

• Center manifolds for partially hyperbolic set without strong unstable connections
• Homoclinic Bifurcations that Give Rise to Heterodimensional Cycles near A Saddle-Focus Equilibrium

Cited by in corpus (4)

• Persistence of Heterodimensional Cycles
• Attractor-repeller collision and the heterodimensional dynamics
• Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles
• Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors