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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, 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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