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

*AR was partially supported by CMUP (UID/MAT/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. AR also acknowledges financial support from Program INVESTIGADOR FCT (IF/00107/2015).

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  • 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.

    Mathematics Subject Classification: Primary: 34C28; 34C37; 37D05; Secondary: 37D45; 37G35.

    Citation:

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  • Figure 1.  Example of a Misiurewicz-type map $ h: {\mathbb{S}}^1 \rightarrow {\mathbb{R}} $. For $ \delta>0 $, the set $ C_{\delta} $ is a neighbourhood of the set of critical points $ C $. These maps are among the simplest examples with non-uniform expansion

    Figure 2.  Illustration of the transition map $ \Psi_{2 \rightarrow 1} $ from $ {\text{Out}}(O_2) $ to $ {\text{In}} (O_1) $. The graph of $ \Phi_2(x, 0) $ may be seen as the first hit of $ W^u(O_2) $ to $ {\text{In}} (O_1) $

    Figure 3.  Graph of $ k(x) = -K_\omega \ln (x) $ and illustration of the sequences $ (\lambda_n)_n $ and $ (\lambda_{(n, a)})_n $ for a fixed $ a \in [0, 2\pi[ $

    Figure 4.  Graph of the map $ h_a $ for $ K_\omega \gg K_\omega^0 $ with $ q = 2 $ (number of critical points). Indicated is a superstable periodic orbit of period 2 as it contains a critical point

    Figure 5.  Adapted bifurcation diagram of [25] for the family (4): 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. In between, rotational horseshoes exist but they may be not observable in numerics; the dynamics is probably dominated either by a sink or bistability between a sink and an attracting torus

    Figure 6.  Illustration (based on plots obtained by using Maple) of $ h_a(x) \pmod{2\pi} $ for $ a = 0.1 $, $ \xi = 0 $ and $ \Phi_2(x, 0) = \sin x $ ($ \Rightarrow q = 2 $). (a) $ K_\omega \approx 0 $ – attracting torus. (b) $ K_\omega = 0.5 $ – attracting torus inside a resonance wedge with a saddle and a sink. (c): $ K_\omega = 2 $ – torus breakdown. (d): $ K_\omega = 5 $ – mixing properties ("big lobe'' of [30])

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