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Dynamics of $ \Gamma_{\mu} .$
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February
2018, 38(2): 941-966.
doi: 10.3934/dcds.2018040
Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors
Dep. de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007, Oviedo, Spain |
For a two parameter family of two-dimensional piecewise linear maps and for every natural number $n$, we prove not only the existence of intervals of parameters for which the respective maps are n times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least $2^n$ strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.
Mathematics Subject Classification:
Primary: 37C70, 37D45; Secondary: 37D25.
Citation:
Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems,
2018, 38
(2)
: 941-966.
doi: 10.3934/dcds.2018040
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References:
| [1] |
M. Benedicks and L. Carleson,
On iterations of $ 1-ax^2\;on\;(-1, 1)$, Annals of Mathematics, 122 (1985), 1-25.
doi: 10.2307/1971367. |
| [2] |
M. Benedicks and L. Carleson,
The dynamics of the Hénon map, Annals of Mathematics, 133 (1991), 73-169.
doi: 10.2307/2944326. |
| [3] |
K. M. Brucks and H. Bruin,
Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617171. |
| [4] |
J. Buzzi,
Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbb{R}$-analytic mappings of the plane, Ergodic Theory and Dynamical Systems, 20 (2000), 697-708.
doi: 10.1017/S0143385700000377. |
| [5] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-642-78043-1. |
| [6] |
W. de Melo,
Renormalization in one-dimensional dynamics, Journal of Difference Equations and Applications, 17 (2011), 1185-1197.
doi: 10.1080/10236190902998016. |
| [7] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
| [8] |
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. |
| [9] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, in Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., 54, Springer, Heidelberg, (2013), 351-366.
doi: 10.1007/978-3-642-38830-9_22. |
| [10] |
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 and Continuous Dynamical Systems -Series B, 19 (2014), 523-541.
doi: 10.3934/dcdsb.2014.19.523. |
| [11] |
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. |
| [12] |
A. Pumariño, J. A. Rodríguez and E. Vigil,
Expanding Baker Maps: Coexistence of strange attractors, Discrete and Continuous Dynamical Systems -Series A, 37 (2017), 1651-1678.
doi: 10.3934/dcds.2017068. |
| [13] |
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. |
| [14] |
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. |
| [15] |
S. Sternberg,
On the structure of local homeomorphisms of euclidean n-space, American Journal of Mathematics, 80 (1958), 623-631.
doi: 10.2307/2372774. |
| [16] |
J. C. Tatjer,
Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.
doi: 10.1017/S0143385701001146. |
show all references
References:
| [1] |
M. Benedicks and L. Carleson,
On iterations of $ 1-ax^2\;on\;(-1, 1)$, Annals of Mathematics, 122 (1985), 1-25.
doi: 10.2307/1971367. |
| [2] |
M. Benedicks and L. Carleson,
The dynamics of the Hénon map, Annals of Mathematics, 133 (1991), 73-169.
doi: 10.2307/2944326. |
| [3] |
K. M. Brucks and H. Bruin,
Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617171. |
| [4] |
J. Buzzi,
Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbb{R}$-analytic mappings of the plane, Ergodic Theory and Dynamical Systems, 20 (2000), 697-708.
doi: 10.1017/S0143385700000377. |
| [5] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-642-78043-1. |
| [6] |
W. de Melo,
Renormalization in one-dimensional dynamics, Journal of Difference Equations and Applications, 17 (2011), 1185-1197.
doi: 10.1080/10236190902998016. |
| [7] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
| [8] |
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. |
| [9] |
A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil,
Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, in Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., 54, Springer, Heidelberg, (2013), 351-366.
doi: 10.1007/978-3-642-38830-9_22. |
| [10] |
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 and Continuous Dynamical Systems -Series B, 19 (2014), 523-541.
doi: 10.3934/dcdsb.2014.19.523. |
| [11] |
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. |
| [12] |
A. Pumariño, J. A. Rodríguez and E. Vigil,
Expanding Baker Maps: Coexistence of strange attractors, Discrete and Continuous Dynamical Systems -Series A, 37 (2017), 1651-1678.
doi: 10.3934/dcds.2017068. |
| [13] |
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. |
| [14] |
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. |
| [15] |
S. Sternberg,
On the structure of local homeomorphisms of euclidean n-space, American Journal of Mathematics, 80 (1958), 623-631.
doi: 10.2307/2372774. |
| [16] |
J. C. Tatjer,
Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.
doi: 10.1017/S0143385701001146. |
Figure 2.
The smoothness domains for a map in $ \mathbb{F} .$
Figure 3.
The set $ \mathcal{R}_1 .$
Figure 4.
The set of parameters $ \mathcal{P}_1 .$
Figure 5.
The set $ \mathcal{R}_{a, b} .$
Figure 6.
The set of parameters $ \mathcal{P}_2 .$
Figure 7.
The set $ \mathcal{R}_{a, b} .$
Figure 9.
The iterates of $ \Delta_0:$ encircled in blue, the triangle $ \Delta_0 ;$ encircled in a dashed blue line, the set $ \Delta_4 .$
Figure 10.
The set of parameters $ \mathcal{P}_{\Delta} .$
Figure 11.
The sets of parameters $ \mathcal{P}_{\Delta}^{\prime} $ (dark grey) and $ \mathcal{P}_{\Delta} $ (pale grey).
Figure 12.
The iterates of $ \Pi_0 :$ encircled in green, the triangle $ \Pi_0 ;$ encircled in a dashed green line, the set $ \Pi_4 .$
Figure 13.
The set of parameters $ \mathcal{P}_{\Pi} .$
Figure 14.
(a) Filled in black, the set $ \mathcal{P}_3 ;$ encircled in a dashed black line, the set $ H_{\Delta}(\mathcal{P}_3) .$ (b) Filled in black, the set $ \mathcal{P}_3 ;$ encircled in a dashed black line, the set $ H_{\Pi}(\mathcal{P}_3) .$
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