Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors

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

“…Hypothesis (P6) corresponds to the partial symmetry-breaking considered in Section 2.4 of [22]. The setting described by (P1)-(P8) generalizes Case (4) of [36].…”

“…In addition, we exhibit a mechanism to construct rotational horseshoes (see §2.2) and strange attractors. This work may be seen as the natural extension of [36], where a two-parameter family of vector fields has been considered to model bifurcations associated to certain unfoldings of Hopf-zero singularities [7,15].…”

“…In this section, we compare and discuss our results with previous works in the literature. Borrowing the arguments used in [36], we may draw, in the first quadrant, two smooth curves, the graphs of t 1 and t 2 drawn in Figure 5, such that:…”

“…(1) for all ω ∈ R + 0 , we have t 2 (ω) ≤ t 1 (w); (2) the region below the graph of t 2 corresponds to flows having an invariant and attracting torus with zero topological entropy (attracting invariant torus); [36] for the family (3.1): below the graph of t 2 , the dynamics is governed by an attracting invariant 2-dimensional torus. Above the graph of t 1 , one observes rotational horseshoes.…”

“…Once the rotational horseshoes develop, they persist and correspond to what the authors of [46] call transient chaos. The rotational horseshoes found in [36] and the associated invariant manifolds are (in general) not observable in numerical simulations because they form a set of Lesbegue measure zero; almost all solutions will be trapped by a sink. As the twisting number K ω gets larger, the deformation on the curve In(O 1 ) ∩ Φ 2 (x, 0) is exaggerated further, giving rise to rank-one attractors created by stretch-and-fold type actions -sustained chaos [46].…”