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Arnold Diffusion in A Priori Chaotic Symplectic Maps

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  • Published: 24 April 2017
  • Volume 353, pages 507–547, (2017)
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Arnold Diffusion in A Priori Chaotic Symplectic Maps
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  • Vassili Gelfreich1 &
  • Dmitry Turaev2,3 

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Abstract

We assume that a symplectic real-analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder. We prove that generically in the real-analytic category the boundaries of the invariant cylinder are connected by trajectories of the map.

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References

  1. Arnold, V.I.: Instability of dynamical systems with many degrees of freedom. Dokl. Akad. Nauk SSSR 156, 9–12 (1964) (Russian)

  2. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Dynamical Systems III, 3rd edn. Encyclopaedia of Mathematical Sciences, vol. 3. Springer, Berlin (2006)

  3. Bernard, P.: Perturbation of a partially hyperbolic Hamiltonian system. C. R. Acad. Sci. Paris Sr. I Math. 323(2), 189–194 (1996) (French)

  4. Bernard P.: The dynamics of pseudographs in convex Hamiltonian systems. J. Am. Math. Soc. 21(3), 615–669 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernard, P.: Arnold’ diffusion: from the a priori unstable to the a priori stable case. In: Proceedings of the International Congress of Mathematicians, Hyderabad, India, vol. III, pp. 1680–1700. Hindustan Book Agency, New Delhi (2010)

  6. Bernard P.: Large normally hyperbolic cylinders in a priori stable Hamiltonian systems. Ann. Henri Poincarè 11(5), 929–942 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Berti M., Biasco L., Bolle P.: Drift in phase space: a new variational mechanism with optimal diffusion time. J. Math. Pures Appl. (9) 82(6), 613–664 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berti M., Bolle P.: Fast Arnold diffusion in systems with three time scales. Discrete Contin. Dyn. Syst. 8(3), 795–811 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bessi U.: Arnold’s diffusion with two resonances. J. Differ. Equ. 137(2), 211–239 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Bolotin S., Treschev D.: Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity 12(2), 365–388 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Bourgain J., Kaloshin V.: On diffusion in high-dimensional Hamiltonian systems. J. Funct. Anal. 229(1), 1–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bounemoura A., Pennamen E.: Instability for a priori unstable Hamiltonian systems: a dynamical approach. Discrete Contin. Dyn. Syst. 32(3), 753–793 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bounemoura A.: Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians. J. Differ. Equ. 249(11), 2905–2920 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Bounemoura A., Marco J.-P.: Improved exponential stability for near-integrable quasi-convex Hamiltonians. Nonlinearity 24(1), 97–112 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Bounemoura, A., Fayad, B., Niederman, L.: Double Exponential Stability for Generic Real-Analytic Elliptic Equilibrium Points (2015). arXiv:1509.00285

  16. Broer H.W., Tangerman F.M.: From a differentiable to a real analytic perturbation theory, applications to the Kupka–Smale theorems. Ergod. Theory Dyn. Syst. 6, 345–362 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Castejon, O., Kaloshin, V.: Random Iteration of Maps on a Cylinder and Diffusive Behavior (2015). arXiv:1501.03319

  18. Chierchia, L., Gallavotti, G.: Drift and diffusion in phase space. Ann. Inst. H. Poincarè Phys. Théor. 60(1),144 (1994) [erratum, Ann. Inst. H. Poincarè Phys. Théor. 68(1),135 (1998)]

  19. Cheng C.-Q., Yan J.: Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differ. Geom. 67(3), 457–517 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cheng C.-Q., Yan J.: Arnold diffusion in Hamiltonian systems: a priori unstable case. J. Differ. Geom. 82(2), 229–277 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cheng, C.-Q.: Arnold Diffusion in Nearly Integrable Hamiltonian Systems, p. 127 (2013) (preprint). arXiv:1207.4016v2

  22. Cresson J.: Symbolic dynamics and Arnold diffusion. J. Differ. Equ. 187(2), 269–292 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Cresson, J., Wiggins, S.: A \({\lambda}\)-Lemma for Normally-Hyperbolic Invariant Manifolds (2005). arXiv:math/0510645 (preprint)

  24. de la Llave, R.: Some recent progress in geometric methods in the instability problem in Hamiltonian mechanics. In: International Congress of Mathematicians, vol. II, pp. 1705–1729. European Mathematical Society, Zurich (2006)

  25. Delshams A., Gelfreich V., Jorba A., Seara T.-M.: Exponentially small splitting of separatrices under fast quasiperiodic forcing. Commun. Math. Phys. 189, 35–71 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Delshams A., Gidea M., Roldán P.: Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete Contin. Dyn. Syst. 33(3), 1089–1112 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Delshams A., Huguet G.: Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems. Nonlinearity 22(8), 1997–2077 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Delshams A., Huguet G.: A geometric mechanism of diffusion: rigorous verification in a priori unstable Hamiltonian systems. J. Differ. Equ. 250(5), 2601–2623 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Delshams A., de la Llave R., Seara T.M.: A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of \({\mathbf{T}^2}\). Commun. Math. Phys. 209(2), 353–392 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Delshams A., de la Llave R., Seara T.M.: Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Adv. Math. 202(1), 64–188 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Delshams A., de la Llave R., Seara T.M.: Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Adv. Math. 202(1), 64–188 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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. Memoirs of the American Mathematical Society, vol. 179, no. 844. American Mathematical Society, Providence, RI (2006)

  33. Delshams, A., de la Llave, R., Seara T.M.: Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation. In: Chreg, W. (ed.) Hamiltonian Dynamical Systems and Applications, pp. 285–336. Springer, Berlin (2008)

  34. Delshams, A., de la Llave, R., Seara, T.M. Instability of high Dimensional Hamiltonian Systems: Multiple Resonances Do Not Impede Diffusion. Adv. Math. 294, 689–755 (2016)

  35. Douady, R.: Stabilité ou instabilité des points fixes elliptiques [Stability or instability of elliptic fixed points]. Ann. Sci. école Norm. Sup. (4) 21(1), 1–46 (1988) (French)

  36. Douady, R., Le Calvez, P.: Example of a non-topologically stable elliptic fixed point in dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 296(21), 895–898 (1983) (French)

  37. Easton R.W., Meiss J.D., Roberts G.: Drift by coupling to an anti-integrable limit. Phys. D 156(3–4), 201–218 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971-1972)

  39. Fermi E.: Beweis, dass ein mechanisches Normalsystem im allgemeinen quasi-ergodisch ist. Phys. Z. 24, 261–265 (1923)

    MATH  Google Scholar 

  40. Fontich E., Martin P.: Arnold diffusion in perturbations of analytic exact symplectic maps. Nonlinear Anal. 42(8), 1397–1412 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Fontich E., Martin P.: Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete Contin. Dynam. Syst. 7(1), 61–84 (2001)

    MathSciNet  MATH  Google Scholar 

  42. Gallavotti, G.: Arnold’s diffusion in isochronous systems. Math. Phys. Anal. Geom. 1(4), 295–312 (1998/1999)

  43. Gallavotti G., Gentile G., Mastropietro V.: Hamilton–Jacobi equation, heteroclinic chains and Arnold diffusion in three time scale systems. Nonlinearity 13(2), 323–334 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Gelfreich V., Turaev D.: Unbounded energy growth in hamiltonian systems with a slowly varying parameter. Commun. Math. Phys. 283(3), 769–794 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Gelfreich V., Turaev D.: Fermi acceleration in non-autonomous billiards. J. Phys. A 41, 212003 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Gidea M., Robinson C.: Shadowing orbits for transition chains of invariant tori alternating with Birkhoff zones of instability. Nonlinearity 20(5), 1115–1143 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Gidea M., Robinson C.: Obstruction argument for transition chains of tori interspersed with gaps. Discrete Contin. Dyn. Syst. Ser. S 2(2), 393–416 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Gidea M., Zgliczynski P.: Covering relations for multidimensional dynamical systems. II. J. Differ. Equ. 202(1), 59–80 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Gidea M., de la Llave R.: Topological methods in the instability problem of Hamiltonian systems. Discrete Contin. Dyn. Syst. 14(2), 295–328 (2006)

    MathSciNet  MATH  Google Scholar 

  50. Gidea, M., de la Llave, R., Seara, T.: A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results (2014). arXiv:1405.0866 (preprint)

  51. Gonchenko, S.V., Turaev, D., Shilnikov, L.: Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20, 241–275 (2007)

  52. Guardia, M., Kaloshin, V.: Orbits of Nearly Integrable Systems Accumulating to KAM Tori (2014). arXiv:1412.7088

  53. Guardia M., Kaloshin V., Zhang J.: A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems. Commun. Math. Phys. 348(1), 321–361 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Guzzo M., Lega E., Froeschlé C.: A numerical study of Arnold diffusion in a priori unstable systems. Commun. Math. Phys. 290(2), 557–576 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Herman, M.-R.: Sur les courbes invariantes par les diffeomorphismes de l’anneau, vol. 1 [On the Curves Invariant Under Diffeomorphisms of the Annulus, vol. 1] Astérisque, pp. 103–104. Soc. Math. de France, Paris (1983) (French)

  56. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583, p. 149. Springer, Berlin (1977)

  57. Holmes P.J., Marsden J.E.: Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys. 23(4), 669–675 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. Jones C.K.R.T., Tin S.-K.: Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete Contin. Dyn. Syst. Ser. S 2(4), 967–1023 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Kaloshin, V., Mather, J., Valdinoci, E.: Instability of resonant totally elliptic points of symplectic maps in dimension 4. Analyse complexe, systemes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II. Astérisque 297, 79–116 (2004)

  60. Kaloshin, V.: Geometric proofs of Mather’s connecting and accelerating theorems. In: Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series No. 310, pp. 81–106. Cambridge University Press, Cambridge (2003)

  61. Kaloshin V., Levi M.: Geometry of Arnold diffusion. SIAM Rev. 50(4), 702–720 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Kaloshin V., Levi M.: An example of Arnold diffusion for near-integrable Hamiltonians. Bull. Am. Math. Soc. 45(3), 409–427 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  63. Kaloshin V., Saprykina M.: An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension. Commun. Math. Phys. 315(3), 643–697 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. Kaloshin, V., Zhang, K.: A Strong Form of Arnold Diffusion for Two and a Half Degrees of Freedom (2012). arXiv:1212.1150 (preprint)

  65. Kaloshin, V., Zhang, K.: A Strong Form of Arnold Diffusion for Three and a Half Degrees of Freedom (2014) (preprint)

  66. Kaloshin, V., Zhang, J., Zhang, K.: Normally Hyperbolic Invariant Laminations and Diffusive Behaviour for the Generalized Arnold Example Away from Resonances (2015). arXiv:1511.04835

  67. Le Calvez P.: Drift for families of twist maps on the annulus. Ergod. Theory Dyn. Syst. 27, 869–879 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  68. Lochak P., Marco J.-P.: Diffusion times and stability exponents for nearly integrable analytic systems. Cent. Eur. J. Math. 3(3), 342–397 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  69. Lochak, P., Marco, J.-P., Sauzin, D.: On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems. Memoirs of the American Mathematical Society, vol. 163, no. 775. American Mathematical Society, Providence, RI (2003)

  70. Marco, J.-P.: Transition along chains of invariant tori for analytic Hamiltonian systems. Ann. Inst. H. Poincaré Phys. Thor. 64(3), 205–252 (1996) (French)

  71. Marco, J.-P.: Arnold diffusion for cusp-generic nearly integrable convex systems on \({\mathbb{A}^3}\) (2016). arXiv:1602.02403 (preprint)

  72. Markus L., Meyer K.R.: Generic Hamiltonian dynamical systems are neither integrable nor ergodic. Mem. AMS 144, 52 (1974)

    MathSciNet  MATH  Google Scholar 

  73. Mather J.N.: Arnold diffusion: announcement of results. J. Math. Sci. 124(5), 5275–5289 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  74. Mather, J.N.: Arnold diffusion by variational methods. In: Pardalos, P.M., Rassias, T. (eds.) Essays in Mathematics and Its Applications, pp. 271–285. Springer, Heidelberg (2012)

  75. Moeckel, R.: Generic drift on Cantor sets of annuli. In: Chenciner, A., Cushman, R., Robinson, C., Xia, Z.J. (eds.) Celestial Mechanics (Evanston, IL, 1999). Contemporary Mathematics, vol. 292, pp. 163–171. American Mathematical Society, Providence (2002)

  76. Moeckel, R.: Transition tori in the five-body problem. J. Differ. Equ. 129, 290–314 (1996)

  77. Nassiri M., Pujals E.R.: Robust transitivity in Hamiltonian dynamics. Ann. Sci. Norm. Sup. (4) 45(2), 191–239 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  78. Nekhoroshev N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Russ. Math. Surv. 32(6), 1–65 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  79. Palis J., de Melo W. A Geometrical Introduction to Dynamical Systems. Springer, Berlin (1982)

  80. Piftankin G.N.: Diffusion speed in the Mather problem. Nonlinearity 19, 2617–2644 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. Piftankin G.N., Treshchev D.V.: Separatrix maps in Hamiltonian systems. Russ. Math. Surv. 62(2), 219–322 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  82. Procesi M.: Exponentially small splitting and Arnold diffusion for multiple time scale systems. Rev. Math. Phys. 15(4), 339–386 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  83. Pugh, C., Shub, M., Wilkinson, A.: Hölder Foliations Revisited, (2011). arXiv:1112.2646 (preprint)

  84. Robinson, C.: Symbolic dynamics for transition tori. In: Chenciner, A., Cushman, R., Robinson, C., Xia, Z.J. (eds.) Celestial Mechanics (Evanston, IL, 1999). Contemporary Mathematics, vol. 292, pp. 199–208. American Mathematical Society, Providence (2002)

  85. Shilnikov, L.P.: On the question of the structure of the neighborhood of a homoclinic tube of an invariant torus. Soviet Math. Dokl. 9, 624–628 (1968)

  86. Shilnikov, L.P., Shilnikov, A., Turaev, D., Chua, L.: Methods of qualitative theory in nonlinear dynamics. Part I. World Scientific Publishing, Singapore (1998)

  87. Treschev D.V.: Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17(5), 1803–1841 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  88. Treschev D.: Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems. Nonlinearity 25(9), 2717–2757 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  89. Tennyson, J.L., Lieberman, M.A., Lichtenberg, A.J.: Diffusion in near-integrable Hamiltonian systems with three degrees of freedom. In: Month, M., Herrera, J.C. (eds.) Nonlinear Dynamics and the Beam–Beam Interaction, vol. 57, pp. 272–301. American Institute of Physics, New York (1979)

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, UK

    Vassili Gelfreich

  2. Imperial College, London, UK

    Dmitry Turaev

  3. Lobachevsky University of Nizhny Novgorod, Nizhny Novgorod, Russia

    Dmitry Turaev

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  1. Vassili Gelfreich
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Correspondence to Dmitry Turaev.

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Communicated by J. Marklof

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Gelfreich, V., Turaev, D. Arnold Diffusion in A Priori Chaotic Symplectic Maps. Commun. Math. Phys. 353, 507–547 (2017). https://doi.org/10.1007/s00220-017-2867-0

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  • Received: 30 March 2015

  • Accepted: 06 January 2017

  • Published: 24 April 2017

  • Issue Date: July 2017

  • DOI: https://doi.org/10.1007/s00220-017-2867-0

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