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Some examples of generalized reflectionless Schrödinger potentials
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 |
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. |
[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. |
[3] |
J. Dieudonné, Foundations of Modern Analysis,, Academic Press, (1969).
|
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[10] |
T. Kato, Perturbation Theory for Linear Operators,, Die Grundlehren der mathematischen Wissenschaften, (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, (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.
doi: 10.1090/S0273-0979-1982-15008-X. |
[13] |
O. Lanford III, Computer assisted proofs,, in Computational methods in field theory (Schladming, (1992), 43.
doi: 10.1007/3-540-55997-3_30. |
[14] |
J. Milnor, On the concept of attractor,, Comm. Math. Phys., 99 (1985), 177.
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.
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).
doi: 10.1142/S0218127413500727. |
[17] |
W. Rudin, Real and Complex Analysis,, 3rd edition, (1987).
|
[18] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
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, (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, (1993).
doi: 10.1007/978-3-642-78043-1. |
[3] |
J. Dieudonné, Foundations of Modern Analysis,, Academic Press, (1969).
|
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[10] |
T. Kato, Perturbation Theory for Linear Operators,, Die Grundlehren der mathematischen Wissenschaften, (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, (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.
doi: 10.1090/S0273-0979-1982-15008-X. |
[13] |
O. Lanford III, Computer assisted proofs,, in Computational methods in field theory (Schladming, (1992), 43.
doi: 10.1007/3-540-55997-3_30. |
[14] |
J. Milnor, On the concept of attractor,, Comm. Math. Phys., 99 (1985), 177.
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.
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).
doi: 10.1142/S0218127413500727. |
[17] |
W. Rudin, Real and Complex Analysis,, 3rd edition, (1987).
|
[18] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
doi: 10.1090/S0002-9904-1967-11798-1. |
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