Renormalizable Expanding Baker Maps: Coexistence of strange attractors

Antonio Pumariño; José Ángel Rodríguez; Enrique Vigil

doi:10.3934/dcds.2017068

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2017, Volume 37, Issue 3: 1651-1678. Doi: 10.3934/dcds.2017068

This issue Previous Article Nonlocal Schrödinger-Kirchhoff equations with external magnetic field Next Article Dynamical canonical systems and their explicit solutions

Renormalizable Expanding Baker Maps: Coexistence of strange attractors

1.
Dep. de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007, Oviedo, Spain
2.
Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687,4169-007 Porto, Portugal

Received: May 31, 2016

Revised: October 31, 2016

Published: May 2017

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Abstract

We introduce the concept of Expanding Baker Maps and renormalizable Expanding Baker Maps in a two-dimensional scenario. For a one-parameter family of Expanding Baker Maps we prove the existence of an interval of parameters for which the respective transformation is renormalizable. Moreover, we show the existence of intervals of parameters for which coexistence of strange attractors takes place.

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Mathematics Subject Classification: Primary:37C70, 37D45;Secondary:37D25.

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Figure 1. (a) A good fold. (b) A not good fold.

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Figure 2. Sequence of good folds.

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Figure 3. An example of EBM.

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Figure 4. Dynamics of $ \Lambda_t .$

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Figure 5. (a) Convex attractor for $\Lambda_t, \ t=0.95.$ (b) Non-simply-connected attractor for $\Lambda_t, \ t=0.765.$

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Figure 6. (a) 8-pieces attractor for $\Lambda_t, \ t=0.74 .$ (b) A magnification of the encircled piece of the previous attractor rotated by an angle $ \pi $.

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Figure 7. (a) Two 32-pieces attractors for $\Lambda_t, \ t=0.715 .$ (b) A magnification of the encircled piece of the previous attractors.

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Figure 8. The $\Lambda_t^8-$invariant domain. Encircled in a dashed blue line, the set $ \widetilde{\Delta}_0 .$

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Figure 9. Dynamic of $ \Gamma_{1, t} .$

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Figure 10. Renormalized attractor for $ t=0.74 .$ (a) Attractor for $\Lambda_t$ (magnification). (b) Attractor for $\Gamma_{1, t}.$

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Figure 11. The smoothness domains for a map in $ \mathbb{F} .$

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Figure 12. The sets $ \mathcal{T}, $ $ \mathcal{T}_{b, 1} $ and $ \mathcal{T}_{b, 2} $.

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Figure 13. The sets $T_{b, 2}, $ $ \mathcal{R}_1 $ and $ \mathcal{R}_{a, b} .$

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Figure 14. The set of parameters $ \mathcal{P}_0 .$ In dashed blue, the curve of parameters for which $ \Gamma_{1, t} $ displays a strictly invariant pentagonal domain.

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Figure 15. The iterates of $ \Delta_0.$ Encircled in a dashed blue line, the set $ \Delta_4 .$

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Figure 16. The iterates of $ \Pi_0 .$ Encircled a dashed green line, the set $ \Pi_4 .$

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Figure 17. The 4-pieces attractors for $ \Gamma_{1, t} $ and the iterates of $ \Delta_0 $ (pale gray or green, in the electronic version) and $ \Pi_0 $ (dark gray or blue, in the electronic version).

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Figure 18. Twice renormalized attractors for $t=0.715 .$ (a) Attractor for $ \Gamma_{2, 1, t} .$ (b) Attractor for $ \Gamma_{2, 2, t} .$

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Figure 19. Three-times renormalized attractors for $\Lambda_t .$ (a) Attractor for $ \Gamma_{3, 1}, \ t=0.71 .$ (b) Attractor for $ \Gamma_{3, 2}, \ t=0.7096 .$ (c) Attractor for $ \Gamma_{3, 3}, \ t=0.7093 .$ (d) Attractor for $ \Gamma_{3, 4}, \ t=0.7087 .$

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