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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

Àlex Haro 1,

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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1080/10236190802332274.  Google Scholar

[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.   Google Scholar

[3]

Kristian Bjerklöv, SNA's in the quasi-periodic quadratic family,, Comm. Math. Phys., 286 (2009), 137.  doi: 10.1007/s00220-008-0626-y.  Google Scholar

[4]

Z. I. Bezhaeva and V. I. Oseledets, On an example of a "strange nonchaotic attractor'',, Funktsional. Anal. i Prilozhen., 30 (1996), 1.   Google Scholar

[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.  doi: 10.3934/dcdsb.2010.14.871.  Google Scholar

[6]

Paul Glendinning, Global attractors of pinched skew products,, Dyn. Syst., 17 (2002), 287.  doi: 10.1080/14689360210160878.  Google Scholar

[7]

Celso Grebogi, Edward Ott, Steven Pelikan and James A. Yorke, Strange attractors that are not chaotic,, Phys. D, 13 (1984), 261.  doi: 10.1016/0167-2789(84)90282-3.  Google Scholar

[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.   Google Scholar

[9]

Àlex Haro and Joaquim Puig, Strange nonchaotic attractors in Harper maps,, Chaos, 16 (2006).   Google Scholar

[10]

À. Haro and C. Simó, To be or not to be an SNA: That is the question, 2005., Available from: \url{http://www.maia.ub.es/dsg/2005/0503haro.pdf}., (2005).   Google Scholar

[11]

Tobias H. Jäger, On the structure of strange non-chaotic attractors in pinched skew products,, Ergodic Theory Dynam. Systems, 27 (2007), 493.   Google Scholar

[12]

Tobias H. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps,, Comm. Math. Phys., 289 (2009), 253.  doi: 10.1007/s00220-009-0753-0.  Google Scholar

[13]

Tobias H. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations,, Mem. Amer. Math. Soc., 201 (2009).   Google Scholar

[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.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[15]

Kunihiko Kaneko, Fractalization of torus,, Progr. Theoret. Phys., 71 (1984), 1112.  doi: 10.1143/PTP.71.1112.  Google Scholar

[16]

Gerhard Keller, A note on strange nonchaotic attractors,, Fund. Math., 151 (1996), 139.   Google Scholar

[17]

Ken-Ichi Inada, On a two-sector model of economic growth: Comments and a generalization,, The Review of Economic Studies, 30 (1963), 119.  doi: 10.2307/2295809.  Google Scholar

[18]

Awadhesh Prasad, Surendra Singh Negi and Ramakrishna Ramaswamy, Strange nonchaotic attractors,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291.  doi: 10.1142/S0218127401002195.  Google Scholar

show all references

References:
[1]

Lluís Alsedà and Michał Misiurewicz, Attractors for unimodal quasiperiodically forced maps,, J. Difference Equ. Appl., 14 (2008), 1175.  doi: 10.1080/10236190802332274.  Google Scholar

[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.   Google Scholar

[3]

Kristian Bjerklöv, SNA's in the quasi-periodic quadratic family,, Comm. Math. Phys., 286 (2009), 137.  doi: 10.1007/s00220-008-0626-y.  Google Scholar

[4]

Z. I. Bezhaeva and V. I. Oseledets, On an example of a "strange nonchaotic attractor'',, Funktsional. Anal. i Prilozhen., 30 (1996), 1.   Google Scholar

[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.  doi: 10.3934/dcdsb.2010.14.871.  Google Scholar

[6]

Paul Glendinning, Global attractors of pinched skew products,, Dyn. Syst., 17 (2002), 287.  doi: 10.1080/14689360210160878.  Google Scholar

[7]

Celso Grebogi, Edward Ott, Steven Pelikan and James A. Yorke, Strange attractors that are not chaotic,, Phys. D, 13 (1984), 261.  doi: 10.1016/0167-2789(84)90282-3.  Google Scholar

[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.   Google Scholar

[9]

Àlex Haro and Joaquim Puig, Strange nonchaotic attractors in Harper maps,, Chaos, 16 (2006).   Google Scholar

[10]

À. Haro and C. Simó, To be or not to be an SNA: That is the question, 2005., Available from: \url{http://www.maia.ub.es/dsg/2005/0503haro.pdf}., (2005).   Google Scholar

[11]

Tobias H. Jäger, On the structure of strange non-chaotic attractors in pinched skew products,, Ergodic Theory Dynam. Systems, 27 (2007), 493.   Google Scholar

[12]

Tobias H. Jäger, Strange non-chaotic attractors in quasiperiodically forced circle maps,, Comm. Math. Phys., 289 (2009), 253.  doi: 10.1007/s00220-009-0753-0.  Google Scholar

[13]

Tobias H. Jäger, The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations,, Mem. Amer. Math. Soc., 201 (2009).   Google Scholar

[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.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[15]

Kunihiko Kaneko, Fractalization of torus,, Progr. Theoret. Phys., 71 (1984), 1112.  doi: 10.1143/PTP.71.1112.  Google Scholar

[16]

Gerhard Keller, A note on strange nonchaotic attractors,, Fund. Math., 151 (1996), 139.   Google Scholar

[17]

Ken-Ichi Inada, On a two-sector model of economic growth: Comments and a generalization,, The Review of Economic Studies, 30 (1963), 119.  doi: 10.2307/2295809.  Google Scholar

[18]

Awadhesh Prasad, Surendra Singh Negi and Ramakrishna Ramaswamy, Strange nonchaotic attractors,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291.  doi: 10.1142/S0218127401002195.  Google Scholar

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