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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 & 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. doi: 10.4007/annals.2006.164.911. Google Scholar

[2]

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

[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, (2016). doi: 10.1007/978-3-319-29662-3. Google Scholar

[4]

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

[5]

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

[6]

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

[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, (2005), 303. doi: 10.1017/CBO9780511526367.006. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

J.-L. Figueras, Fiberwise Hyperbolic Invariant Tori in Quasiperiodically Skew Product Systems,, PhD thesis, (2011). Google Scholar

[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. doi: 10.1137/100809222. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

A. Haro and R. de la Llave, Spectral theory and dynamical systems,, , (). Google Scholar

[18]

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

[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. doi: 10.3934/dcdsb.2006.6.1261. Google Scholar

[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. doi: 10.1016/j.jde.2005.10.005. Google Scholar

[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. doi: 10.1137/050637327. Google Scholar

[22]

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

[23]

A. Haro and C. Simó, To be or not to be a SNA: That is the question,, Preprint, (2005). Google Scholar

[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. doi: 10.1007/BF02564647. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

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

[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. doi: 10.1142/S0218127407019780. Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[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. doi: 10.1016/0022-0396(74)90067-9. Google Scholar

[36]

R. J. Sacker and G. R. Sell, A spectral theory for linear almost periodic differential equations,, In International Conference on Differential Equations (Proc., (1974), 698. Google Scholar

[37]

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

[38]

J. Stark, Invariant graphs for forced systems,, Phys. D, 109 (1997), 163. doi: 10.1016/S0167-2789(97)00167-X. Google Scholar

[39]

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

[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. doi: 10.1134/S1560354711010060. Google Scholar

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. doi: 10.4007/annals.2006.164.911. Google Scholar

[2]

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

[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, (2016). doi: 10.1007/978-3-319-29662-3. Google Scholar

[4]

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

[5]

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

[6]

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

[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, (2005), 303. doi: 10.1017/CBO9780511526367.006. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

J.-L. Figueras, Fiberwise Hyperbolic Invariant Tori in Quasiperiodically Skew Product Systems,, PhD thesis, (2011). Google Scholar

[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. doi: 10.1137/100809222. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

A. Haro and R. de la Llave, Spectral theory and dynamical systems,, , (). Google Scholar

[18]

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

[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. doi: 10.3934/dcdsb.2006.6.1261. Google Scholar

[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. doi: 10.1016/j.jde.2005.10.005. Google Scholar

[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. doi: 10.1137/050637327. Google Scholar

[22]

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

[23]

A. Haro and C. Simó, To be or not to be a SNA: That is the question,, Preprint, (2005). Google Scholar

[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. doi: 10.1007/BF02564647. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

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

[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. doi: 10.1142/S0218127407019780. Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[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. doi: 10.1016/0022-0396(74)90067-9. Google Scholar

[36]

R. J. Sacker and G. R. Sell, A spectral theory for linear almost periodic differential equations,, In International Conference on Differential Equations (Proc., (1974), 698. Google Scholar

[37]

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

[38]

J. Stark, Invariant graphs for forced systems,, Phys. D, 109 (1997), 163. doi: 10.1016/S0167-2789(97)00167-X. Google Scholar

[39]

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

[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. doi: 10.1134/S1560354711010060. Google Scholar

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