Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors

Abstract

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.

Persistence of Heterodimensional Cycles

“…In a series of papers [25][26][27] we have shown that heterodimensional cycles can be a part of a pseudohyperbolic chain-transitive attractor which appears in systems with Shilnikov loops [18,32,38] or after a periodic perturbation of the Lorenz attractor [39]. It follows from our results in the current paper that the attractor in such systems remains heterodimensional for an open set of parameter values.…”

“…By construction, these estimates are uniform for all systems C 2 -close to f , and when the system is at least C 3 -smooth, the same estimates are also true for the first derivatives with respect to parameters ε. We note also that in the coordinates (27) we have the local stable manifold of O 1 and the image of the local unstable manifold of O 2 straightened:…”

Homoclinic tangencies leading to robust heterodimensional cycles

“…In the codimension one case we can obtain additional conclusions, compare with [35,Lemmas 11 and 12]. Lemma 5.4.…”

“…Next lemma about expansion of volume of local submanifolds tangent to the cone field C cu in Remark 4.1 by the return maps R k is a variation of [35,Lemma 5]. To state it recall the definition of J P = J P (f ) in (1.2) and that a submanifold S is tangent to C cu if T X S ⊂ C cu (X) for every X ∈ S. Proof.…”

“…In this paper, we focus on item (a), providing settings where homoclinic tangencies associated to saddles lead to C 1 robust cycles after small C r perturbations 2 , r 1. Our constructions are motivated by the ones in [35], where heterodimensional cycles are obtained from homoclinic tangencies with some symmetry conditions. To describe our setting, we first comment some global aspects and restrictions concerning Question 2.…”