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Rank-one strange attractors versus heteroclinic tangles

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 two different mechanisms for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. The unfolding of each parameter is associated to the emergence of different dynamical scenarios (rank-one attractors and heteroclinic tangles). These relative positions may be determined using the Melnikov method.

    Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "lar-ge" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and non-uniformly hyperbolic attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a prevalent phenomenon.

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

    Citation:

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  • Figure 1.  The dynamics of $ (2.1) $ defined $ \mathcal{V}\subset {\mathbb{R}}^2 $ is governed by the existence of an heteroclinic cycle associated to $ O_1 $ and $ O_2 $. $ \ell_1, \ell_2 $: heteroclinic connections; $ \mathcal{V}^\star $: inner basin of attraction of the cycle (absorbing domain); $ \mathcal{A}: $ region limited by the cycle

    Figure 2.  (a) Scheme of the effects of the parameters $ \mu_1, \mu_2 $ on the equations (2.3). (b) Sketch of the local and transition maps. $ I, II, III $ and $ IV $ represent cross sections $ {\text{Out}}( \textbf{C}_1) $, $ {\text{In}}( \textbf{C}_2) $, $ {\text{Out}}( \textbf{C}_2) $ and $ {\text{In}}( \textbf{C}_1) $, respectively

    Figure 3.  Plausible bifurcation diagram associated to an element $ f_{(\mu_1, \mu_2)} $ of the family $ \mathfrak{X}_{\Gamma}^4(\mathbb{V}) $. I – the flow has an invariant two-dimensional torus if $ \omega\approx 0 $ and a rank-one strange attractor if $ \omega \gg 1 $. II – transition region; III – heteroclinic tangle; Hom – curve that corresponds to the emergence of a homoclinic tangency associated to $ \textbf{C}_1 $

    Figure 4.  Example of a Misiurewicz-type map $ h_a: {\mathbb{S}}^1 \rightarrow {\mathbb{R}} $. For $ \delta>0 $, the set $ C_{\delta} $ is a $ \delta $-neighbourhood of the set of critical points $ C $

    Figure 5.  Transition maps from $ {\text{Out}}( \textbf{C}_1) $ to $ {\text{In}} ( \textbf{C}_2) $ (Case I) and from $ {\text{Out}}( \textbf{C}_2) $ to $ {\text{In}} ( \textbf{C}_1) $ (Case II). In Case I: $ W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2) = \emptyset $. In Case II: $ W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1) $. Double bars mean that the sides are identified

    Figure 6.  Graph of the map $ h_a $ for $ a = \mu_n $ and $ \omega \gg 1 $ with $ q = 2 $ (number of critical points). Indicated is a superstable periodic orbit of period 2

    Figure 7.  The set $ \eta_u = W^u( \textbf{C}_2)\cap {\text{In}}^+ ( \textbf{C}_1) $ is mapped by $ {Loc}_2\circ \Psi_{1\rightarrow 2}\circ Loc_1 $ into a spiral accumulating on the circle defined by $ \overline{y}_1^{(2)} = 0 $ (ie the parametrisation of $ {\text{Out}}( \textbf{C}_2)\cap W^u( \textbf{C}_2) $). There is a sequence $ (\mu_i)_i $ for which the flow of $ \mathcal{G}_{\mu_i} $ exhibits a quadratic heteroclinic tangency

    Figure 8.  Four types of configuration for (2.2) when $ \mu_1>0 $ and $ \mu_2 = 0 $. (a), (b): one contractible periodic solution; (c): two contractible periodic solutions; (d) one non-contractible periodic solution. Double bars mean that the sides are identified

    Table 1.  Four generic unfoldings for the dynamics of (2.3)

    Configuration $ W^u( \textbf{C}_1) \pitchfork W^s( \textbf{C}_2) $ $ W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2)=\emptyset $
    $ W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1) $ Case 1 Case 2
    $ W^u( \textbf{C}_2) \cap W^s( \textbf{C}_1)=\emptyset $ Case 3 Case 4
     | Show Table
    DownLoad: CSV

    Table 2.  Coordinates and notation of the cross sections after the change of coordinates (6.1) and (6.2)

    Set Notation Set Notation
    $ {\text{In}} ( \textbf{C}_1) $ $ \left(y_1^{(1)}, y_2^{(1)}, \theta^{(1)}\right) $ $ {\text{In}} ( \textbf{C}_2) $ $ \left(y_1^{(2)}, y_2^{(2)}, \theta^{(2)}\right) $
    $ {\text{Out}} ( \textbf{C}_1) $ $ \left(\overline{y}_1^{(1)}, \overline{y}_2^{(1)}, \overline{\theta}^{(1)}\right) $ $ {\text{Out}} ( \textbf{C}_2) $ $ \left(\overline{y}_1^{(2)}, \overline{y}_2^{(2)}, \overline{\theta}^{(2)}\right) $
     | Show Table
    DownLoad: CSV

    Table 3.  Notation for the next sections where $ \mu_1, \mu_2\in\, \, ]0, \varepsilon] $ with $ \varepsilon $ is small

    $ \mathcal{F}_{(\mu_1, 0)} \mapsto \mathcal{F}_{\mu} $ Sections 7 and 8
    $ \mathcal{G}_{(0, \mu_2)}\mapsto \mathcal{G}_{\mu} $ Sections 9 and 10
     | Show Table
    DownLoad: CSV

    Table 4.  Overview of the results in the literature and the contribution of the present article (in blue) for the dynamics of (2.3)

    Configuration $ W^u( \textbf{C}_1) \pitchfork W^s( \textbf{C}_2) $ $ W^u( \textbf{C}_1) \cap W^s( \textbf{C}_2)=\emptyset $
    $ W^u( \textbf{C}_2) \pitchfork W^s( \textbf{C}_1) $ Case 1
    Heteroclinic tangles
    (Horseshoes, tangencies, sinks
    Newhouse phenomena, pulses)
    [5,10,18]
    Case 2 (Novelty of this article)
    ($ \mu_1>0 $ and $ \mu_2=0 $)
    Region with torus if $ \omega \approx 0 $
    Region with rank-one attractors if $ \omega \gg 1 $
    Superstable sinks if $ \omega \gg 1 $
    ($ \mu_1=0 $ and $ \mu_2>0 $)
    Non-uniformly hyperbolic strange attractors
    if $ \omega \gg 1 $
    ($ \mu_1, \mu_2>0 $)
    Heteroclinic tangles prevail
    $ W^u( \textbf{C}_2) \cap W^s( \textbf{C}_1)=\emptyset $ Case 3
    Similar to Case 2
    Case 4
    Region with torus if $ \omega \approx 0 $
    Region with rank-one attractors if $ \omega \gg 1 $
    [12]
     | Show Table
    DownLoad: CSV

    Table 5.  Notation

    Notation Definition/meaning Section
    $ \mathcal{V} $ Open region of $ {\mathbb{R}}^2 $ where equation (2.1) is well defined § 2.1
    $ O_1, O_2 $ Saddle-equilibria of the equation (2.1) § 2.1
    $ {\ell}_1, {\ell}_2 $ Connections from $ O_1 $ to $ O_2 $ and from $ O_2 $ to $ O_1 $ § 2.1
    $ \mathcal{A} $ Region limited by the heteroclinic cycle $ {\ell}_1\cup {\ell}_2 $ § 2.1
    $ \mathcal{V}^\star $ Inner basin of attraction of the heteroclinic cycle $ {\ell}_1\cup {\ell}_2 $
    (absorbing domain)
    § 2.1
    $ \mathbb{V} $ $ \mathcal{V}\times {\mathbb{S}}^1 $ – open region where equation (2.3) is defined § 2.2
    $ \textbf{C}_1, \textbf{C}_2 $ Saddle periodic solutions of the equation (2.3) § 2.2
    $ \Gamma $ Heteroclinic cycle associated to $ \textbf{C}_1, \textbf{C}_2 $ § 2.2
    $ \mathbb{A} $ $ \mathcal{A}\times {\mathbb{S}}^1 $ § 2.2
    $ \mathbb{V}^\star $ $ \mathcal{V}^\star\times {\mathbb{S}}^1 $ § 2.2
    $ \mathcal{L}_1, \mathcal{L}_2 $ Connections from $ \textbf{C}_1 $ to $ \textbf{C}_2 $ and from $ \textbf{C}_2 $ to $ \textbf{C}_1 $ § 2.2
    $ V_1, V_2 $ Hollow cylinders around $ \textbf{C}_1 $ and $ \textbf{C}_2 $ § 2.2
    $ A\equiv_\mathbb{V^\star} B $ The manifolds $ A $ and $ B $ coincide within $ \mathbb{V}^\star $ § 2.3
    $ \mathcal{F}_{(\mu_1, \mu_2)}\equiv \mathcal{F}_{\mu} $ First return map to $ {\text{Out}}( \textbf{C}_1) $ § 6.5 and
    Table 3
    $ \mathcal{G}_{(\mu_1, \mu_2)}\equiv \mathcal{G}_{\mu} $ First return map to $ {\text{Out}}( \textbf{C}_2) $ § 6.5 and
    Table 3
     | Show Table
    DownLoad: CSV
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