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

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