Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors

We prove that heterodimensional cycles can be created by unfolding a pair of homoclinic tangencies in a certain class of C ^r-diffeomorphisms (r=3, ⋯, ∞,ω). This implies the existence of a C ²-open domain in the space of dynamical systems with a certain type of symmetry where systems with heterodimensional cycles are dense in C ^r. In particular, we describe a class of three-dimensional flows with a Lorenz-like attractor such that an arbitrarily small time-periodic 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.