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August  2016, 9(4): 1095-1107. doi: 10.3934/dcdss.2016043

A note on the fractalization of saddle invariant curves in quasiperiodic systems

1. 

Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala

2. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona

Received  November 2015 Revised  April 2016 Published  August 2016

The purpose of this paper is to describe a new mechanism of destruction of saddle invariant curves in quasiperiodically forced systems, in which an invariant curve experiments a process of fractalization, that is, the curve gets increasingly wrinkled until it breaks down. The phenomenon resembles the one described for attracting invariant curves in a number of quasiperiodically forced dissipative systems, and that has received the attention in the literature for its connections with the so-called Strange Non-Chaotic Attractors. We present a general conceptual framework that provides a simple unifying mathematical picture for fractalization routes in dissipative and conservative systems.
Citation: Jordi-Lluís Figueras, Àlex Haro. A note on the fractalization of saddle invariant curves in quasiperiodic systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1095-1107. doi: 10.3934/dcdss.2016043
References:
[1]

A. Avila and R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. (2), 164 (2006), 911-940. doi: 10.4007/annals.2006.164.911.

[2]

H. Broer, G. Huitema, F. Takens and B. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. Am. Math. Soc., 83 (1990), viii+175 pp. doi: 10.1090/memo/0421.

[3]

M. Canadell, J.-L. Figueras, A. Haro, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations, Applied Mathematical Sciences. Springer-Verlag, New York, 2016. doi: 10.1007/978-3-319-29662-3.

[4]

M. Canadell and A. Haro, A KAM-like theorem for quasi-periodic normally hyperbolic invariant tori. Preprint, 2015.

[5]

C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, volume 70 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/surv/070.

[6]

C. Chicone and R. C. Swanson, Spectral theory for linearizations of dynamical systems, J. Differential Equations, 40 (1981), 155-167. doi: 10.1016/0022-0396(81)90015-2.

[7]

M.-C. Ciocci, A. Litvak-Hinenzon and H. Broer, Survey on dissipative {KAM} theory including quasi-periodic bifurcation theory), In Geometric mechanics and symmetry, volume 306 of London Math. Soc. Lecture Note Ser., pages 303-355. Cambridge Univ. Press, Cambridge, 2005. doi: 10.1017/CBO9780511526367.006.

[8]

S. Datta, R. Ramaswamy and A. Prasad, Fractalization route to strange nonchaotic dynamics, Phys. Rev. E, 70 (2004), 046203. doi: 10.1103/PhysRevE.70.046203.

[9]

E. I. Dinaburg and J. G. Sinaĭ, The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. i Priložen., 9 (1975), 8-21.

[10]

L. H. Eliasson, Floquet solutions for the $1$-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482. doi: 10.1007/BF02097013.

[11]

J.-L. Figueras, Fiberwise Hyperbolic Invariant Tori in Quasiperiodically Skew Product Systems, PhD thesis, Universitat de Barcelona, Barcelona, Spain, May 2011.

[12]

J.-L. Figueras and A. Haro, Reliable computation of robust response tori on the verge of breakdown, SIAM J. Appl. Dyn. Syst., 11 (2012), 597-628. doi: 10.1137/100809222.

[13]

J.-L. Figueras and À. Haro, Triple collisions of invariant bundles, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2069-2082. doi: 10.3934/dcdsb.2013.18.2069.

[14]

J.-L. Figueras and À. Haro., Different scenarions for hyperbolicity in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16 pp. doi: 10.1063/1.4938185.

[15]

C. Grebogi, E. Ott, S. Pelikan and J. Yorke, Strange attractors that are not chaotic, Phys. D, 13 (1984), 261-268. doi: 10.1016/0167-2789(84)90282-3.

[16]

J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.

[17]

A. Haro and R. de la Llave, Spectral theory and dynamical systems, ftp://ftp.ma.utexas.edu/pub/papers/llave/spectral.pdf.

[18]

A. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8pp. doi: 10.1063/1.2150947.

[19]

A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300 (electronic). doi: 10.3934/dcdsb.2006.6.1261.

[20]

A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005.

[21]

A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207 (electronic). doi: 10.1137/050637327.

[22]

À. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7pp. doi: 10.1063/1.2259821.

[23]

A. Haro and C. Simó, To be or not to be a SNA: That is the question, Preprint, http://www.maia.ub.es/dsg/2005/index.html, 2005.

[24]

M.-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. doi: 10.1007/BF02564647.

[25]

R. Johnson, Analyticity of spectral subbundles, J. Differential Equations, 35 (1980), 366-387. doi: 10.1016/0022-0396(80)90034-0.

[26]

R. Johnson and G. Sell, Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems, J. Differential Equations, 41 (1981), 262-288. doi: 10.1016/0022-0396(81)90062-0.

[27]

R. A. Johnson, The Oseledec and Sacker-Sell spectra for almost periodic linear systems: An example, Proc. Amer. Math. Soc., 99 (1987), 261-267. doi: 10.1090/S0002-9939-1987-0870782-7.

[28]

À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X.

[29]

A. 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.

[30]

À. Jorba, J. C. Tatjer, C. Núñez and R. Obaya, Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3895-3928. doi: 10.1142/S0218127407019780.

[31]

K. Kaneko, Fractalization of torus, Progr. Theoret. Phys., 71 (1984), 1112-1115. doi: 10.1143/PTP.71.1112.

[32]

K. Kaneko, Collapse of Tori and Genesis of Chaos in Dissipative Systems, World Scientific Publishing Co., Singapore, 1986. doi: 10.1142/0175.

[33]

J. N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.

[34]

T. Nishikawa and K. Kaneko., Fractalization of a torus as a strange nonchaotic attractor, Phys. Rev. E, 54 (1996), 6114-6124. doi: 10.1103/PhysRevE.54.6114.

[35]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.

[36]

R. J. Sacker and G. R. Sell, A spectral theory for linear almost periodic differential equations, In International Conference on Differential Equations (Proc., Univ. Southern California, Los Angeles, Calif., 1974), pages 698-708. Academic Press, New York, 1975.

[37]

O. Sosnovtseva, U. Feudel, J. Kurths and A. Pikovsky, Multiband strange nonchaotic attractors in quasiperiodically forced systems, Physics Letters A, 218 (1996), 255-267. doi: 10.1016/0375-9601(96)00399-4.

[38]

J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179, Physics and dynamics between chaos, order, and noise (Berlin, 1996). doi: 10.1016/S0167-2789(97)00167-X.

[39]

J. Stark, Regularity of invariant graphs for forced systems, Ergodic Theory Dynam. Systems, 19 (1999), 155-199. doi: 10.1017/S0143385799126555.

[40]

R. Vitolo, H. Broer and C. Simó, Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems, Regul. Chaotic Dyn., 16 (2011), 154-184. doi: 10.1134/S1560354711010060.

show all references

References:
[1]

A. Avila and R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. (2), 164 (2006), 911-940. doi: 10.4007/annals.2006.164.911.

[2]

H. Broer, G. Huitema, F. Takens and B. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. Am. Math. Soc., 83 (1990), viii+175 pp. doi: 10.1090/memo/0421.

[3]

M. Canadell, J.-L. Figueras, A. Haro, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations, Applied Mathematical Sciences. Springer-Verlag, New York, 2016. doi: 10.1007/978-3-319-29662-3.

[4]

M. Canadell and A. Haro, A KAM-like theorem for quasi-periodic normally hyperbolic invariant tori. Preprint, 2015.

[5]

C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, volume 70 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/surv/070.

[6]

C. Chicone and R. C. Swanson, Spectral theory for linearizations of dynamical systems, J. Differential Equations, 40 (1981), 155-167. doi: 10.1016/0022-0396(81)90015-2.

[7]

M.-C. Ciocci, A. Litvak-Hinenzon and H. Broer, Survey on dissipative {KAM} theory including quasi-periodic bifurcation theory), In Geometric mechanics and symmetry, volume 306 of London Math. Soc. Lecture Note Ser., pages 303-355. Cambridge Univ. Press, Cambridge, 2005. doi: 10.1017/CBO9780511526367.006.

[8]

S. Datta, R. Ramaswamy and A. Prasad, Fractalization route to strange nonchaotic dynamics, Phys. Rev. E, 70 (2004), 046203. doi: 10.1103/PhysRevE.70.046203.

[9]

E. I. Dinaburg and J. G. Sinaĭ, The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. i Priložen., 9 (1975), 8-21.

[10]

L. H. Eliasson, Floquet solutions for the $1$-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482. doi: 10.1007/BF02097013.

[11]

J.-L. Figueras, Fiberwise Hyperbolic Invariant Tori in Quasiperiodically Skew Product Systems, PhD thesis, Universitat de Barcelona, Barcelona, Spain, May 2011.

[12]

J.-L. Figueras and A. Haro, Reliable computation of robust response tori on the verge of breakdown, SIAM J. Appl. Dyn. Syst., 11 (2012), 597-628. doi: 10.1137/100809222.

[13]

J.-L. Figueras and À. Haro, Triple collisions of invariant bundles, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2069-2082. doi: 10.3934/dcdsb.2013.18.2069.

[14]

J.-L. Figueras and À. Haro., Different scenarions for hyperbolicity in quasiperiodic area preserving twist maps, Chaos, 25 (2015), 123119, 16 pp. doi: 10.1063/1.4938185.

[15]

C. Grebogi, E. Ott, S. Pelikan and J. Yorke, Strange attractors that are not chaotic, Phys. D, 13 (1984), 261-268. doi: 10.1016/0167-2789(84)90282-3.

[16]

J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201. doi: 10.1063/1.524170.

[17]

A. Haro and R. de la Llave, Spectral theory and dynamical systems, ftp://ftp.ma.utexas.edu/pub/papers/llave/spectral.pdf.

[18]

A. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8pp. doi: 10.1063/1.2150947.

[19]

A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300 (electronic). doi: 10.3934/dcdsb.2006.6.1261.

[20]

A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228 (2006), 530-579. doi: 10.1016/j.jde.2005.10.005.

[21]

A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207 (electronic). doi: 10.1137/050637327.

[22]

À. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7pp. doi: 10.1063/1.2259821.

[23]

A. Haro and C. Simó, To be or not to be a SNA: That is the question, Preprint, http://www.maia.ub.es/dsg/2005/index.html, 2005.

[24]

M.-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. doi: 10.1007/BF02564647.

[25]

R. Johnson, Analyticity of spectral subbundles, J. Differential Equations, 35 (1980), 366-387. doi: 10.1016/0022-0396(80)90034-0.

[26]

R. Johnson and G. Sell, Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems, J. Differential Equations, 41 (1981), 262-288. doi: 10.1016/0022-0396(81)90062-0.

[27]

R. A. Johnson, The Oseledec and Sacker-Sell spectra for almost periodic linear systems: An example, Proc. Amer. Math. Soc., 99 (1987), 261-267. doi: 10.1090/S0002-9939-1987-0870782-7.

[28]

À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X.

[29]

A. 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.

[30]

À. Jorba, J. C. Tatjer, C. Núñez and R. Obaya, Old and new results on strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3895-3928. doi: 10.1142/S0218127407019780.

[31]

K. Kaneko, Fractalization of torus, Progr. Theoret. Phys., 71 (1984), 1112-1115. doi: 10.1143/PTP.71.1112.

[32]

K. Kaneko, Collapse of Tori and Genesis of Chaos in Dissipative Systems, World Scientific Publishing Co., Singapore, 1986. doi: 10.1142/0175.

[33]

J. N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.

[34]

T. Nishikawa and K. Kaneko., Fractalization of a torus as a strange nonchaotic attractor, Phys. Rev. E, 54 (1996), 6114-6124. doi: 10.1103/PhysRevE.54.6114.

[35]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.

[36]

R. J. Sacker and G. R. Sell, A spectral theory for linear almost periodic differential equations, In International Conference on Differential Equations (Proc., Univ. Southern California, Los Angeles, Calif., 1974), pages 698-708. Academic Press, New York, 1975.

[37]

O. Sosnovtseva, U. Feudel, J. Kurths and A. Pikovsky, Multiband strange nonchaotic attractors in quasiperiodically forced systems, Physics Letters A, 218 (1996), 255-267. doi: 10.1016/0375-9601(96)00399-4.

[38]

J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179, Physics and dynamics between chaos, order, and noise (Berlin, 1996). doi: 10.1016/S0167-2789(97)00167-X.

[39]

J. Stark, Regularity of invariant graphs for forced systems, Ergodic Theory Dynam. Systems, 19 (1999), 155-199. doi: 10.1017/S0143385799126555.

[40]

R. Vitolo, H. Broer and C. Simó, Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems, Regul. Chaotic Dyn., 16 (2011), 154-184. doi: 10.1134/S1560354711010060.

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