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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,
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 .$
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