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On strange attractors in a class of pinched skew products
1. | Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona |
  Since the dynamics on the strange attractor is the one given by the base homeomorphism, we will say that it is a strange chaotic attractor or a strange non-chaotic attractor depending on the fact that the dynamics on the base is chaotic or non-chaotic. The results complement the paper by G. Keller on rigorous proofs of existence of strange non-chaotic attractors.
References:
[1] |
Lluís Alsedà and Michał Misiurewicz, Attractors for unimodal quasiperiodically forced maps, J. Difference Equ. Appl., 14 (2008), 1175-1196.
doi: 10.1080/10236190802332274. |
[2] |
Kristian Bjerklöv, Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations, Ergodic Theory Dynam. Systems, 25 (2005), 1015-1045. |
[3] |
Kristian Bjerklöv, SNA's in the quasi-periodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161.
doi: 10.1007/s00220-008-0626-y. |
[4] |
Z. I. Bezhaeva and V. I. Oseledets, On an example of a "strange nonchaotic attractor'', Funktsional. Anal. i Prilozhen., 30 (1996), 1-9, 95. |
[5] |
Henk W. Broer, Carles Simó and Renato Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905.
doi: 10.3934/dcdsb.2010.14.871. |
[6] |
Paul Glendinning, Global attractors of pinched skew products, Dyn. Syst., 17 (2002), 287-294.
doi: 10.1080/14689360210160878. |
[7] |
Celso Grebogi, Edward Ott, Steven Pelikan and James A. Yorke, Strange attractors that are not chaotic, Phys. D, 13 (1984), 261-268.
doi: 10.1016/0167-2789(84)90282-3. |
[8] |
Michael-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502. |
[9] |
Àlex Haro and Joaquim Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7 pp. |
[10] |
À. Haro and C. Simó, To be or not to be an SNA: That is the question, 2005. Available from: http://www.maia.ub.es/dsg/2005/0503haro.pdf. |
[11] |
Tobias H. Jäger, On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510. |
[12] |
Tobias H. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys., 289 (2009), 253-289.
doi: 10.1007/s00220-009-0753-0. |
[13] |
Tobias H. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc., 201 (2009), vi+106 pp. |
[14] |
Àngel Jorba and Joan Carles 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. |
[15] |
Kunihiko Kaneko, Fractalization of torus, Progr. Theoret. Phys., 71 (1984), 1112-1115.
doi: 10.1143/PTP.71.1112. |
[16] |
Gerhard Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148. |
[17] |
Ken-Ichi Inada, On a two-sector model of economic growth: Comments and a generalization, The Review of Economic Studies, 30 (1963), 119-127.
doi: 10.2307/2295809. |
[18] |
Awadhesh Prasad, Surendra Singh Negi and Ramakrishna Ramaswamy, Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291-309.
doi: 10.1142/S0218127401002195. |
show all references
References:
[1] |
Lluís Alsedà and Michał Misiurewicz, Attractors for unimodal quasiperiodically forced maps, J. Difference Equ. Appl., 14 (2008), 1175-1196.
doi: 10.1080/10236190802332274. |
[2] |
Kristian Bjerklöv, Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations, Ergodic Theory Dynam. Systems, 25 (2005), 1015-1045. |
[3] |
Kristian Bjerklöv, SNA's in the quasi-periodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161.
doi: 10.1007/s00220-008-0626-y. |
[4] |
Z. I. Bezhaeva and V. I. Oseledets, On an example of a "strange nonchaotic attractor'', Funktsional. Anal. i Prilozhen., 30 (1996), 1-9, 95. |
[5] |
Henk W. Broer, Carles Simó and Renato Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905.
doi: 10.3934/dcdsb.2010.14.871. |
[6] |
Paul Glendinning, Global attractors of pinched skew products, Dyn. Syst., 17 (2002), 287-294.
doi: 10.1080/14689360210160878. |
[7] |
Celso Grebogi, Edward Ott, Steven Pelikan and James A. Yorke, Strange attractors that are not chaotic, Phys. D, 13 (1984), 261-268.
doi: 10.1016/0167-2789(84)90282-3. |
[8] |
Michael-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502. |
[9] |
Àlex Haro and Joaquim Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7 pp. |
[10] |
À. Haro and C. Simó, To be or not to be an SNA: That is the question, 2005. Available from: http://www.maia.ub.es/dsg/2005/0503haro.pdf. |
[11] |
Tobias H. Jäger, On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510. |
[12] |
Tobias H. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps, Comm. Math. Phys., 289 (2009), 253-289.
doi: 10.1007/s00220-009-0753-0. |
[13] |
Tobias H. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations, Mem. Amer. Math. Soc., 201 (2009), vi+106 pp. |
[14] |
Àngel Jorba and Joan Carles 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. |
[15] |
Kunihiko Kaneko, Fractalization of torus, Progr. Theoret. Phys., 71 (1984), 1112-1115.
doi: 10.1143/PTP.71.1112. |
[16] |
Gerhard Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148. |
[17] |
Ken-Ichi Inada, On a two-sector model of economic growth: Comments and a generalization, The Review of Economic Studies, 30 (1963), 119-127.
doi: 10.2307/2295809. |
[18] |
Awadhesh Prasad, Surendra Singh Negi and Ramakrishna Ramaswamy, Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291-309.
doi: 10.1142/S0218127401002195. |
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