On strange attractors in a class of pinched skew products

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February 2012, 32(2): 605-617. doi: 10.3934/dcds.2012.32.605

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

Received October 2010 Revised July 2011 Published September 2011

Abstract
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In this paper we construct strange attractors in a class of pinched skew product dynamical systems over homeomorphims on a compact metric space. We assume that maps between fibers satisfy Inada conditions and that the base space is a super-repeller (it is invariant and its vertical Lyapunov exponent is $+\infty$). In particular, we prove the existence of a measurable but non-continuous invariant graph, whose vertical Lyapunov exponent is negative. %We will refer to such an object as a strange attractor.
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.

Keywords: pinched skew products., Strange attractors, strange non-chaotic attractors.

Mathematics Subject Classification: Primary: 37C60, 37C70; Secondary: 37B5.

Citation: Àlex Haro. On strange attractors in a class of pinched skew products. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 605-617. doi: 10.3934/dcds.2012.32.605

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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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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