"Large" strange attractors in the unfolding of a heteroclinic attractor

Abstract

<p style='text-indent:20px;'>We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.</p>

Abundance of infinite switching

“…Our findings have the same flavour to those of [30], even though Hypothesis (H7) is different and the classical theory of [32] does not hold in the case under consideration. As far as we know, there is no analogue of the effect of singularities of the singular limit in previous studies about infinite switching.…”

“…Item (1) of Lemma 7.1 says that the derivative of h a at x ∈ S 1 \{0, π} goes like the inverse of the distance of x ∈ S 1 \S to the singular set S: if x approaches S, then h ′ a (x) explodes. This does happen for Misiurewicz-type maps (see §5 of [30]). Now, we define an interval around the critical point c ∈ C which "generates" a strange attractor.…”

“…The proof of Proposition 3.2 is adapted from [30] and will be revisited in Section 5. Since δ > 1, for λ > 0 small enough, the second component of G λ is contracting and its dynamics is dominated by the family of circle maps with singularities:…”

“…In this section, we compute the singular limit set associated to G λ defined in Proposition 3.2. The formal definition of singular limit may be found in [30,32]. 7 Notice that r ≥ 3 is the class of differentiability of the initial vector field 6.1.…”