Advertisement

Annales Henri Poincaré

, Volume 15, Issue 1, pp 29–60 | Cite as

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

  • Daniel BlazevskiEmail author
  • Rafael de la Llave
Article

Abstract

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un)stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et al. (J Diff Equ, 2012), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show that the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.

Keywords

Manifold Invariant Manifold Unstable Manifold Stable Manifold Decay Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnol’d V.I.: Instability of dynamical systems with many degrees of freedom. Dokl. Akad. Nauk SSSR. 156, 9–12 (1964)MathSciNetGoogle Scholar
  2. 2.
    Bilki B., Erdubak M., Mungan M., Weisskopf Y.: Structure formation of a later of adatoms of a quasicrystaline substrate: molecular dynamics study. Phys. Rev. B. 75(045437), 1–6 (2007)Google Scholar
  3. 3.
    Braun O.M., Kivshar Y.S.: onlinear dynamics of the Frenkel–Kontorova model. Phys. Rep. 306(1–2), 108 (1998)MathSciNetGoogle Scholar
  4. 4.
    Braun, O.M., Kivshar, Y.S.: The Frenkel–Kontorova model. Texts and Monographs in Physics. Concepts, methods, and applications. Springer, Berlin (2004)Google Scholar
  5. 5.
    Bunimovich L.A., Sinai Y.G.: Spacetime chaos in coupled map lattices. Nonlinearity 1(4), 491–516 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Coombes, S., Bressloff, P.C. (eds.): Bursting: the Genesis of Rhythm in the Nervous System. World Scientific Publishing Co. Pte. Ltd., Singapore (2005)Google Scholar
  7. 7.
    Chazottes, J.-R., Fernandez, B. (eds.): Dynamics of coupled map lattices and of related spatially extended systems. Lectures delivered at the school-forum CML 2004, Paris, France, June 21–July 2, 2004. Lecture Notes in Physics 671. Springer, Berlin (2005)Google Scholar
  8. 8.
    Cabré X., Fontich E., de la Llave R.: The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J. 52(2), 283–328 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cabré X., Fontich E., de la Llave R.: The parameterization method for invariant manifolds. III. Overview and applications. J. Differ. Eq. 218(2), 444–515 (2005)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Delshams A., de la Llave R., Seara T.M.: A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Mem. Am. Math. Soc. 179(844), viii+141 (2006)Google Scholar
  11. 11.
    Dauxois T., Peyrard M., Willis C.R.: Localized breather-like solution in a discrete Klein–Gordon model and application to DNA. Phys. D 57(3–4), 267–282 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds.): Dynamics and thermodynamics of systems with long-range interactions. Lecture Notes in Physics, vol. 602. Springer, Berlin, 2002. Lectures from the conference held in Les Houches, February 18–22 (2002)Google Scholar
  13. 13.
    Du Toit P., Mezić I., Marsden J.: Coupled oscillator models with no scale separation. Phys. D. 238(5), 490–501 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ermentrout, G.B., Terman, D.H.: Mathematical foundations of neuroscience. Interdisciplinary applied mathematics, vol. 35. Springer, New YorkGoogle Scholar
  15. 15.
    Floria, L.M., Baesens, C., Gómez-Gardeñes, J.: The Frenkel–Kontorova model [7]Google Scholar
  16. 16.
    Fontich E., de la Llave R., Martín P.: Dynamical systems on lattices with decaying interaction I: a functional analysis framework. J. Differ. Eq. 250(6), 2838–2886 (2011)ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    Fontich E., de la Llave R., Martín P.: Dynamical systems on lattices with decaying interaction II: hyperbolic sets and their invariant manifolds. J. Differ. Eq. 250(6), 2887–2926 (2011)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Fontich E., de la Llave R., Sire Y.: Construction of invariant whiskered tori by a parameterization method. I. Maps and flows in finite dimensions. J. Differ. Eq. 246(8), 3136–3213 (2009)ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Fontich, E., de la Llave, R., Sire, Y.: Construction of invariant whiskered tori by a parameterization method. part II: quasi-periodic and almost periodic breathers in coupled map lattices. J. Diff. Equ. (2012, submitted)Google Scholar
  20. 20.
    Friesecke G., Pego R.L.: Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12(6), 1601–1627 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems. i. In: Newell, A.C. (ed.) Nonlinear wave motion. Lectures in applied mathematics, vol. 15, pp. 143–156. Am. Math. Soc., Providence (1955)Google Scholar
  22. 22.
    Gallavotti, G. (ed.): The Fermi–Pasta–Ulam problem. Lecture Notes in Physics, vol. 728. A status report. Springer, Berlin (2008)Google Scholar
  23. 23.
    Gerstner, W.W., Kistler, W. M.: Spiking neuron models. Single neurons, populations, plasticity. Cambridge University Press, Cambridge (2002)Google Scholar
  24. 24.
    Haro A., de la Llave R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differ. Equ. 228(2), 530–579 (2006)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Izhikevich E.M.: Dynamical systems in neuroscience: the geometry of excitability and bursting. Computational Neuroscience. MIT Press, Cambridge (2007)Google Scholar
  26. 26.
    Jiang M., de la Llave R.: Smooth dependence of thermodynamic limits of SRB-measures. Comm. Math. Phys. 211, 303–333 (2000). doi: 10.1007/s002200050814 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Jiang M., Pesin Y.B.: Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations. Comm. Math. Phys. 193(3), 675–711 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kaneko, K. (ed.): Theory and applications of coupled map lattices. nonlinear science: theory and applications. John Wiley & Sons Ltd, Chichester (1993)Google Scholar
  29. 29.
    Kaneko K., Bagley R.J.: Arnold diffusion, ergodicity and intermittency in a coupled standard mapping. Phys. Lett. A. 110(9), 435–440 (1985)ADSCrossRefMathSciNetGoogle Scholar
  30. 30.
    Mather J.N.: Characterization of Anosov diffeomorphisms. Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math. 30, 479–483 (1968)MathSciNetGoogle Scholar
  31. 31.
    Peyrard M.: Nonlinear dynamics and statistical physics of DNA. Nonlinearity 17(2), R1–R40 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Peyrard M., Sire Y.: Breathers in biomolecules? Conference “Energy Localisation and transfer in Crystals, Biomolecules and Josephson Arrays”. Adv. Ser. Nonlinear Dyn. 22(2), 391–418 (2004)Google Scholar
  33. 33.
    Pesin Ya.B., Yurchenko A.A.: Some physical models described by the reaction-diffusion equation, and coupled map lattices. Uspekhi Mat. Nauk. 59(3(357)), 81–114 (2004)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Rudin, W.: Functional analysis. In: International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc., New YorkGoogle Scholar
  35. 35.
    Rugh H.H.: Coupled maps and analytic function spaces. Ann. Sci. École Norm. Sup. (4) 35(4), 489–535 (2002)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Susuki Y., Mezić I., Hikihara T.: Coherent swing instability of power grids. J. Nonlinear Sci. 21(3), 403–439 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mechanical and Process EngineeringInstitute for Mechanical Systems, ETH ZurichZurichSwitzerland
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations