Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors

TL;DR

This paper demonstrates how heterodimensional cycles can be generated through perturbations in certain symmetric dynamical systems, leading to dense occurrences within a specific class of flows with Lorenz-like attractors.

Contribution

It establishes the creation of heterodimensional cycles via unfolding homoclinic tangencies and describes a class of flows where such cycles are dense.

Findings

Heterodimensional cycles can be created by unfolding homoclinic tangencies.

Dense heterodimensional cycles exist in a C2-open domain of symmetric systems.

Small perturbations of Lorenz-like flows can produce heterodimensional cycles within chain-transitive attractors.

Abstract

We prove that heterodimensional cycles can be created by unfolding a pair of homoclinic tangencies in a certain class of C-infinity diffeomorphisms. This implies the existence of a C2- open domain in the space of dynamical systems with a certain type of symmetry where systems with heterodimensional cycles are dense in C-infinity. In particular, we describe a class of three-dimensional flows with a Lorenz-like attractor such that an arbitrarily small perturbation of any such flow can belong to this domain - in this case the corresponding heterodimensional cycles belong to a chain-transitive attractor of the perturbed flow.