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

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

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.
Citation: Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Local study of a renormalization operator for 1D maps under quasiperiodic forcing. Discrete and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[17]

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

[18]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[17]

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

[18]

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

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