Figure 1.
(a) A good fold. (b) A not good fold.
-
Previous Article
Dynamical canonical systems and their explicit solutions
- DCDS Home
- This Issue
-
Next Article
Nonlocal Schrödinger-Kirchhoff equations with external magnetic field
March
2017, 37(3): 1651-1678.
doi: 10.3934/dcds.2017068
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 |
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.
Mathematics Subject Classification:
Primary:37C70, 37D45;Secondary:37D25.
Citation:
Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalizable Expanding Baker Maps: Coexistence of strange attractors. Discrete & Continuous Dynamical Systems - A,
2017, 37
(3)
: 1651-1678.
doi: 10.3934/dcds.2017068
![]()
References:
| [1] |
V. Baladi, D. Rockmore, N. Tongring and C. Tresser,
Renormalization on the n-dimensional torus, Nonlinearity, 5 (1992), 1111-1136.
doi: 10.1088/0951-7715/5/5/005. |
| [2] |
M. Benedicks and L. Carleson,
On iterations of $ 1-ax^2 $ on $(-1, 1)$, Ann. Math., 122 (1985), 1-25.
doi: 10.2307/1971367. |
| [3] |
M. Benedicks and L. Carleson,
The dynamics of the Hénon map, Ann. Math., 133 (1991), 73-169.
doi: 10.2307/2944326. |
| [4] |
K. M. Brucks and H. Bruin,
Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617171. |
| [5] |
E. Colli,
Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.
doi: 10.1016/S0294-1449(98)80001-2. |
| [6] |
J. M. Gambaudo, S. van Strein and C. Tresser,
Henon-like maps with strange attractors: There exist $ C^{∞} $
Kupka-Smale diffeomorphisms on $ S^2 $
with neither sinks nor sources, Nonlinearity, 2 (1989), 287-304.
doi: 10.1088/0951-7715/2/2/005. |
| [7] |
M. Lyubich and M. Martens,
Renormalization of Hénon Maps: A survey, Dynamics, Games and Science. I, Springer Proc. Math., 1 (2011), 597-618.
doi: 10.1007/978-3-642-11456-4_37. |
| [8] |
W. de Melo,
Renormalization in one-dimensional dynamics, J. Difference Equ. Appl., 17 (2011), 1185-1197.
doi: 10.1080/10236190902998016. |
| [9] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-642-78043-1. |
| [10] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
| [11] |
S. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
| [12] |
J. Palis and J. C. Yoccoz,
Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math., 172 (1994), 91-136.
doi: 10.1007/BF02392792. |
| [13] |
A. Pumariño and J. A. Rodríguez,
Coexistence and Persistence of Strange Attractors Lecture Notes in Mathematics, 1658, Springer-Verlag, Berlin, 1997.
doi: 10.1007/BFb0093337. |
| [14] |
A. Pumariño and J. A. Rodríguez,
Coexistence and persistence of infinitely many strange attractors, Ergod. Theory Dyn. Syst., 21 (2001), 1511-1523.
doi: 10.1017/S0143385701001730. |
| [15] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, Progress and Challenges in Dynamical Systems, Springer, 54 (2013), 351-366.
doi: 10.1007/978-3-642-38830-9_22. |
| [16] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 523-541.
doi: 10.3934/dcdsb.2014.19.523. |
| [17] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Chaotic dynamics for 2-d tent maps, Nonlinearity, 28 (2015), 407-434.
doi: 10.1088/0951-7715/28/2/407. |
| [18] |
A. Pumariño and J. C. Tatjer,
Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms, Nonlinearity, 19 (2006), 2833-2852.
doi: 10.1088/0951-7715/19/12/006. |
| [19] |
A. Pumariño and J. C. Tatjer,
Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism, Discrete and Continuous Dynamical Systems, Series B, 8 (2007), 971-1005.
doi: 10.3934/dcdsb.2007.8.971. |
| [20] |
J. C. Tatjer,
Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.
doi: 10.1017/S0143385701001146. |
| [21] |
J. A. Yorke and K. T. Alligood,
Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull. A.M.S., 9 (1983), 319-322.
doi: 10.1090/S0273-0979-1983-15191-1. |
show all references
References:
| [1] |
V. Baladi, D. Rockmore, N. Tongring and C. Tresser,
Renormalization on the n-dimensional torus, Nonlinearity, 5 (1992), 1111-1136.
doi: 10.1088/0951-7715/5/5/005. |
| [2] |
M. Benedicks and L. Carleson,
On iterations of $ 1-ax^2 $ on $(-1, 1)$, Ann. Math., 122 (1985), 1-25.
doi: 10.2307/1971367. |
| [3] |
M. Benedicks and L. Carleson,
The dynamics of the Hénon map, Ann. Math., 133 (1991), 73-169.
doi: 10.2307/2944326. |
| [4] |
K. M. Brucks and H. Bruin,
Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617171. |
| [5] |
E. Colli,
Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.
doi: 10.1016/S0294-1449(98)80001-2. |
| [6] |
J. M. Gambaudo, S. van Strein and C. Tresser,
Henon-like maps with strange attractors: There exist $ C^{∞} $
Kupka-Smale diffeomorphisms on $ S^2 $
with neither sinks nor sources, Nonlinearity, 2 (1989), 287-304.
doi: 10.1088/0951-7715/2/2/005. |
| [7] |
M. Lyubich and M. Martens,
Renormalization of Hénon Maps: A survey, Dynamics, Games and Science. I, Springer Proc. Math., 1 (2011), 597-618.
doi: 10.1007/978-3-642-11456-4_37. |
| [8] |
W. de Melo,
Renormalization in one-dimensional dynamics, J. Difference Equ. Appl., 17 (2011), 1185-1197.
doi: 10.1080/10236190902998016. |
| [9] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-642-78043-1. |
| [10] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
| [11] |
S. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
| [12] |
J. Palis and J. C. Yoccoz,
Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math., 172 (1994), 91-136.
doi: 10.1007/BF02392792. |
| [13] |
A. Pumariño and J. A. Rodríguez,
Coexistence and Persistence of Strange Attractors Lecture Notes in Mathematics, 1658, Springer-Verlag, Berlin, 1997.
doi: 10.1007/BFb0093337. |
| [14] |
A. Pumariño and J. A. Rodríguez,
Coexistence and persistence of infinitely many strange attractors, Ergod. Theory Dyn. Syst., 21 (2001), 1511-1523.
doi: 10.1017/S0143385701001730. |
| [15] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, Progress and Challenges in Dynamical Systems, Springer, 54 (2013), 351-366.
doi: 10.1007/978-3-642-38830-9_22. |
| [16] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 523-541.
doi: 10.3934/dcdsb.2014.19.523. |
| [17] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Chaotic dynamics for 2-d tent maps, Nonlinearity, 28 (2015), 407-434.
doi: 10.1088/0951-7715/28/2/407. |
| [18] |
A. Pumariño and J. C. Tatjer,
Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms, Nonlinearity, 19 (2006), 2833-2852.
doi: 10.1088/0951-7715/19/12/006. |
| [19] |
A. Pumariño and J. C. Tatjer,
Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism, Discrete and Continuous Dynamical Systems, Series B, 8 (2007), 971-1005.
doi: 10.3934/dcdsb.2007.8.971. |
| [20] |
J. C. Tatjer,
Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.
doi: 10.1017/S0143385701001146. |
| [21] |
J. A. Yorke and K. T. Alligood,
Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull. A.M.S., 9 (1983), 319-322.
doi: 10.1090/S0273-0979-1983-15191-1. |
Figure 2.
Sequence of good folds.
Figure 3.
An example of EBM.
Figure 4.
Dynamics of $ \Lambda_t .$
Figure 5.
(a) Convex attractor for $\Lambda_t, \ t=0.95.$ (b) Non-simply-connected attractor for $\Lambda_t, \ t=0.765.$
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 $ .
Figure 7.
(a) Two 32-pieces attractors for $\Lambda_t, \ t=0.715 .$ (b) A magnification of the encircled piece of the previous attractors.
Figure 8.
The $\Lambda_t^8-$ invariant domain. Encircled in a dashed blue line, the set $ \widetilde{\Delta}_0 .$
Figure 9.
Dynamic of $ \Gamma_{1, t} .$
Figure 10.
Renormalized attractor for $ t=0.74 .$ (a) Attractor for $\Lambda_t$ (magnification). (b) Attractor for $\Gamma_{1, t}.$
Figure 11.
The smoothness domains for a map in $ \mathbb{F} .$
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.
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).
Figure 18.
Twice renormalized attractors for $t=0.715 .$ (a) Attractor for $ \Gamma_{2, 1, t} .$ (b) Attractor for $ \Gamma_{2, 2, t} .$
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 .$
| [1] |
Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 941-966. doi: 10.3934/dcds.2018040 |
| [2] |
Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533 |
| [3] |
Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 |
| [4] |
Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651 |
| [5] |
Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979 |
| [6] |
Claudio Buzzi, Claudio Pessoa, Joan Torregrosa. Piecewise linear perturbations of a linear center. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3915-3936. doi: 10.3934/dcds.2013.33.3915 |
| [7] |
Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106 |
| [8] |
Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 657-670. doi: 10.3934/dcdsb.2018201 |
| [9] |
Ciprian D. Coman. Dissipative effects in piecewise linear dynamics. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 163-177. doi: 10.3934/dcdsb.2003.3.163 |
| [10] |
Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91 |
| [11] |
Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105 |
| [12] |
Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685 |
| [13] |
Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451 |
| [14] |
Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753 |
| [15] |
Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917 |
| [16] |
Laura Poggiolini, Marco Spadini. Local inversion of a class of piecewise regular maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2207-2224. doi: 10.3934/cpaa.2018105 |
| [17] |
Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839 |
| [18] |
Peter Giesl, Sigurdur Hafstein. Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3539-3565. doi: 10.3934/dcds.2012.32.3539 |
| [19] |
Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943 |
| [20] |
Makoto Mori. Higher order mixing property of piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 915-934. doi: 10.3934/dcds.2000.6.915 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]