American Institute of Mathematical Sciences

Advanced
August  2016, 9(4): 1171-1188. doi: 10.3934/dcdss.2016047

Local study of a renormalization operator for 1D maps under quasiperiodic forcing

Àngel Jorba 1, , Pau Rabassa 2, and  Joan Carles Tatjer 3,

1. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585 , 08007 Barcelona
2. 

School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
3. 

Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Received  September 2015 Revised  December 2015 Published  August 2016

The authors have recently introduced an extension of the classical one dimensional (doubling) renormalization operator to the case where the one dimensional map is forced quasiperiodically. In the classic case the dynamics around the fixed point of the operator is key for understanding the bifurcations of one parameter families of one dimensional unimodal maps. Here we perform a similar study of the (linearised) dynamics around the fixed point for further application to quasiperiodically forced unimodal maps.
Keywords: discretization., Quasiperiodically forced maps, renormalization.
Mathematics Subject Classification: Primary: 37C55; Secondary: 37E20, 37G3.
Citation: Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Local study of a renormalization operator for 1D maps under quasiperiodic forcing. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1171-1188. doi: 10.3934/dcdss.2016047
References:
[1]

H. Broer and F. Takens, Dynamical Systems and Chaos, vol. 172 of Applied Mathematical Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-6870-8.  Google Scholar

[2]

W. de Melo and S. van Strien, One-dimensional Dynamics, vol. 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[3]

J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1969, Enlarged and corrected printing, Pure and Applied Mathematics, Vol. 10-I.  Google Scholar

[4]

R. Fabbri, T. Jäger, R. Johnson and G. Keller, A Sharkovskii-type theorem for minimally forced interval maps, Topol. Methods Nonlinear Anal., 26 (2005), 163-188. doi: 10.12775/TMNA.2005.029.  Google Scholar

[5]

U. Feudel, S. Kuznetsov and A. Pikovsky, Strange Nonchaotic Attractors, vol. 56 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.  Google Scholar

[6]

À. Jorba, C. Núñez, R. Obaya and J. C. Tatjer, Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3895-3928. doi: 10.1142/S0218127407019780.  Google Scholar

[7]

À. Jorba, P. Rabassa and J. C. Tatjer, A renormalization operator for 1D maps under quasi-periodic perturbations, Nonlinearity, 28 (2015), 1017-1042. doi: 10.1088/0951-7715/28/4/1017.  Google Scholar

[8]

À. Jorba, P. Rabassa and J. C. Tatjer, Period doubling and reducibility in the quasi-periodically forced logistic map, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1507-1535. doi: 10.3934/dcdsb.2012.17.1507.  Google Scholar

[9]

À. Jorba and J. C. Tatjer, A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567. doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[11]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[12]

O. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434. doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[13]

O. Lanford III, Computer assisted proofs, in Computational methods in field theory (Schladming, 1992), vol. 409 of Lecture Notes in Phys., Springer, Berlin, 1992, 43-58. doi: 10.1007/3-540-55997-3_30.  Google Scholar

[14]

J. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.  Google Scholar

[15]

A. Prasad, S. Negi and R. Ramaswamy, Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291-309. doi: 10.1142/S0218127401002195.  Google Scholar

[16]

P. Rabassa, À. Jorba and J. C. Tatjer, A numerical study of universality and self-similarity in some families of forced logistic maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350072, 11pp. doi: 10.1142/S0218127413500727.  Google Scholar

[17]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[18]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

show all references

References:
[1]

H. Broer and F. Takens, Dynamical Systems and Chaos, vol. 172 of Applied Mathematical Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-6870-8.  Google Scholar

[2]

W. de Melo and S. van Strien, One-dimensional Dynamics, vol. 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[3]

J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1969, Enlarged and corrected printing, Pure and Applied Mathematics, Vol. 10-I.  Google Scholar

[4]

R. Fabbri, T. Jäger, R. Johnson and G. Keller, A Sharkovskii-type theorem for minimally forced interval maps, Topol. Methods Nonlinear Anal., 26 (2005), 163-188. doi: 10.12775/TMNA.2005.029.  Google Scholar

[5]

U. Feudel, S. Kuznetsov and A. Pikovsky, Strange Nonchaotic Attractors, vol. 56 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.  Google Scholar

[6]

À. Jorba, C. Núñez, R. Obaya and J. C. Tatjer, Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3895-3928. doi: 10.1142/S0218127407019780.  Google Scholar

[7]

À. Jorba, P. Rabassa and J. C. Tatjer, A renormalization operator for 1D maps under quasi-periodic perturbations, Nonlinearity, 28 (2015), 1017-1042. doi: 10.1088/0951-7715/28/4/1017.  Google Scholar

[8]

À. Jorba, P. Rabassa and J. C. Tatjer, Period doubling and reducibility in the quasi-periodically forced logistic map, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1507-1535. doi: 10.3934/dcdsb.2012.17.1507.  Google Scholar

[9]

À. Jorba and J. C. Tatjer, A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567. doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar

[11]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[12]

O. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434. doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[13]

O. Lanford III, Computer assisted proofs, in Computational methods in field theory (Schladming, 1992), vol. 409 of Lecture Notes in Phys., Springer, Berlin, 1992, 43-58. doi: 10.1007/3-540-55997-3_30.  Google Scholar

[14]

J. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280.  Google Scholar

[15]

A. Prasad, S. Negi and R. Ramaswamy, Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291-309. doi: 10.1142/S0218127401002195.  Google Scholar

[16]

P. Rabassa, À. Jorba and J. C. Tatjer, A numerical study of universality and self-similarity in some families of forced logistic maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350072, 11pp. doi: 10.1142/S0218127413500727.  Google Scholar

[17]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[18]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[1]

Tingting Zhang, Àngel Jorba, Jianguo Si. Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6599-6622. doi: 10.3934/dcds.2016086
[2]

Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651
[3]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[4]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429
[5]

Max Gunzburger, Sung-Dae Yang, Wenxiang Zhu. Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 569-587. doi: 10.3934/dcdsb.2007.8.569
[6]

Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106
[7]

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
[8]

Àngel Jorba, Joan Carles Tatjer. A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 537-567. doi: 10.3934/dcdsb.2008.10.537
[9]

João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641
[10]

Frank D. Grosshans, Jürgen Scheurle, Sebastian Walcher. Invariant sets forced by symmetry. Journal of Geometric Mechanics, 2012, 4 (3) : 271-296. doi: 10.3934/jgm.2012.4.271
[11]

Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861
[12]

Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423
[13]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437
[14]

Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
[15]

Corrado Falcolini, Laura Tedeschini-Lalli. A numerical renormalization method for quasi–conservative periodic attractors. Journal of Computational Dynamics, 2020, 7 (2) : 461-468. doi: 10.3934/jcd.2020018
[16]

Ning Sun, Shaoyun Shi, Wenlei Li. Singular renormalization group approach to SIS problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3577-3596. doi: 10.3934/dcdsb.2020073
[17]

Richard Evan Schwartz. Outer billiards on the Penrose kite: Compactification and renormalization. Journal of Modern Dynamics, 2011, 5 (3) : 473-581. doi: 10.3934/jmd.2011.5.473
[18]

Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633
[19]

Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477
[20]

Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (111)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy