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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, (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, (1993).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[3]

J. Dieudonné, Foundations of Modern Analysis,, Academic Press, (1969).   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.  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., (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.  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.  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.  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.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators,, Die Grundlehren der mathematischen Wissenschaften, (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, (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.  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), 43.  doi: 10.1007/3-540-55997-3_30.  Google Scholar

[14]

J. Milnor, On the concept of attractor,, Comm. Math. Phys., 99 (1985), 177.  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.  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).  doi: 10.1142/S0218127413500727.  Google Scholar

[17]

W. Rudin, Real and Complex Analysis,, 3rd edition, (1987).   Google Scholar

[18]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  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, (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, (1993).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[3]

J. Dieudonné, Foundations of Modern Analysis,, Academic Press, (1969).   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.  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., (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.  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.  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.  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.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators,, Die Grundlehren der mathematischen Wissenschaften, (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, (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.  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), 43.  doi: 10.1007/3-540-55997-3_30.  Google Scholar

[14]

J. Milnor, On the concept of attractor,, Comm. Math. Phys., 99 (1985), 177.  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.  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).  doi: 10.1142/S0218127413500727.  Google Scholar

[17]

W. Rudin, Real and Complex Analysis,, 3rd edition, (1987).   Google Scholar

[18]

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

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