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Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

  • Amadeu Delshams
  • Marian Gidea
  • Rafael de la Llave
  • Tere M. Seara
Part of the NATO Science for Peace and Security Series book series (NAPSB)

We present (informally) some geometric structures that imply instability in Hamiltonian systems. We also present some finite calculations which can establish the presence of these structures in a given near integrable systems or in systems for which good numerical information is available. We also discuss some quantitative features of the diffusion mechanisms such as time of diffusion, Hausdorff dimension of diffusing orbits, etc.

Keywords

Hamiltonian System Invariant Manifold Unstable Manifold Implicit Function Theorem Homoclinic Orbit 
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.

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References

  1. [AA67]
    V.I. Arnold and A. Avez. Ergodic problems of classical mechanics. Benjamin, New York, 1967.Google Scholar
  2. [AKN88]
    V.I. Arnold, V.V. Kozlov, and A.I. Neishtadt. Dynamical Systems III, volume 3 of Encyclopaedia Math. Sci. Springer, Berlin, 1988.Google Scholar
  3. [Ale68a]
    V. M. Alekseev. Quasirandom dynamical systems. I. Quasirandom diffeomorphisms. Mat. Sb. (N.S.), 76 (118):72-134, 1968.Google Scholar
  4. [Ale68b]
    V. M. Alekseev. Quasirandom dynamical systems. II. One-dimensional nonlinear vibrations in a periodically perturbed field. Mat. Sb. (N.S.), 77 (119):545-601, 1968.MathSciNetGoogle Scholar
  5. [Ale69]
    V. M. Alekseev. Quasirandom dynamical systems. III. Quasirandom vibrations of one-dimensional oscillators. Mat. Sb. (N.S.), 78 (120):3-50, 1969.MathSciNetGoogle Scholar
  6. [Ale81]
    V.M. Alekseev. Quasirandom oscillations and qualitative questions in celestial mechanics. Transl., Ser. 2, Am. Math. Soc., 116:97-169, 1981.Google Scholar
  7. [AM78]
    R. Abraham and J. E. Marsden. Foundations of mechanics. Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, MA, 1978. Second edition, revised and enlarged, With the assistance of Tudor Ratiu and Richard Cushman.Google Scholar
  8. [Ang87]
    Sigurd Angenent. The shadowing lemma for elliptic PDE. In Dynamics of infinite-dimensional systems (Lisbon, 1986), pages 7-22. Springer, Berlin, 1987.Google Scholar
  9. [AO98]
    J. Miguel Alonso and R. Ortega. Roots of unity and unbounded motions of an asym-metric oscillator. J. Differential Equations, 143(1):201-220, 1998.MATHMathSciNetGoogle Scholar
  10. [Arn63]
    V. I. Arnol’d. Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Math. Surveys, 18(6):85-191, 1963.MATHGoogle Scholar
  11. [Arn64]
    V.I. Arnold. Instability of dynamical systems with several degrees of freedom. Sov. Math. Doklady, 5:581-585, 1964.Google Scholar
  12. [Arn89]
    V. I. Arnold. Mathematical methods of classical mechanics. Springer, New York, second edition, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein.Google Scholar
  13. [Ban88]
    V. Bangert. Mather sets for twist maps and geodesics on tori. In Dynamics reported, Vol. 1, pages 1-56. Teubner, Stuttgart, 1988.Google Scholar
  14. [BBB03]
    M. Berti, L. Biasco, and P. Bolle. Drift in phase space: a new variational mechanism with optimal diffusion time. J. Math. Pures Appl. (9), 82(6):613-664, 2003.MATHMathSciNetGoogle Scholar
  15. [BCV01]
    U. Bessi, L. Chierchia, and E. Valdinoci. Upper bounds on Arnold diffusion times via Mather theory. J. Math. Pures Appl. (9), 80(1):105-129, 2001.MATHMathSciNetGoogle Scholar
  16. [Bes96]
    U. Bessi. An approach to Arnold’s diffusion through the calculus of variations. Nonlinear Anal., 26(6):1115-1135, 1996.MATHMathSciNetGoogle Scholar
  17. [BK94]
    I. U. Bronstein and A. Ya. Kopanskiı. Smooth invariant manifolds and normal forms, volume 7 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. World Sci. Publ., River Edge, NJ, 1994.Google Scholar
  18. [BK05]
    J. Bourgain and V. Kaloshin. On diffusion in high-dimensional Hamiltonian systems. J. Funct. Anal., 229(1):1-61, 2005.MATHMathSciNetGoogle Scholar
  19. [BLZ98]
    P. W. Bates, K. Lu, and C. Zeng. Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. Amer. Math. Soc., 135(645):viii+129, 1998.MathSciNetGoogle Scholar
  20. [BP01]
    L. Barreira and Ya. Pesin. Lectures on Lyapunov exponents and smooth ergodic theory. In Smooth ergodic theory and its applications (Seattle, WA, 1999), pages 3-106. Amer. Math. Soc., Providence, RI, 2001. Appendix A by M. Brin and Appendix B by D. Dolgopyat, H. Hu and Pesin.Google Scholar
  21. [BT79]
    M. Breitenecker and W. Thirring. Scattering theory in classical dynamics. Riv. Nuovo Cimento (3), 2(4):21, 1979.MathSciNetGoogle Scholar
  22. [BT99]
    S. Bolotin and D. Treschev. Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity, 12(2):365-388, 1999.MATHMathSciNetGoogle Scholar
  23. [Car81]
    J. R. Cary. Lie transform perturbation theory for Hamiltonian systems. Phys. Rep., 79(2):129-159, 1981.MathSciNetGoogle Scholar
  24. [CDMR06]
    E. Canalias, A. Delshams, J. Masdemont, and P. Roldán. The scattering map in the planar restricted three body problem. Celestial Mech. Dynam. Astronom., 95(1-4):155-171, 2006.MATHGoogle Scholar
  25. [CG98]
    L. Chierchia and G. Gallavotti. Drift and diffusion in phase space. Ann. Inst. H. Poincaré Phys. Théor., 60(1):1-144, 1994. Erratum, Ann. Inst. H. Poincaré, Phys. Théor. 68 (1):135, 1998.MATHMathSciNetGoogle Scholar
  26. [CG03]
    Jacky Cresson and Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete Contin. Dyn. Syst., 9(2):451-470, 2003.MATHMathSciNetGoogle Scholar
  27. [Chi79]
    B.V. Chirikov. A universal instability of many-dimensional oscillator systems. Phys. Rep., 52(5):264-379, 1979.MathSciNetGoogle Scholar
  28. [CI99]
    G. Contreras and R. Iturriaga. Global minimizers of autonomous Lagrangians. 22o Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium]. In-stituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.Google Scholar
  29. [Con68]
    C. C. Conley. Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math., 16:732-746, 1968.MATHMathSciNetGoogle Scholar
  30. [Con78]
    C. Conley. Isolated invariant sets and the Morse index, volume 38 of CBMS Regional Conference Series in Mathematics. Amer. Math. Soc., Providence, RI, 1978.Google Scholar
  31. [Cre97]
    J. Cresson. A λ -lemma for partially hyperbolic tori and the obstruction property. Lett. Math. Phys., 42(4):363-377, 1997.MATHMathSciNetGoogle Scholar
  32. [CY04a]
    C.-Q. Cheng and J. Yan. Arnold diffusion in Hamiltonian systems: 1 a priori unstable case. Preprint 04-265, mp_arc@math.utexas.edu, 2004.Google Scholar
  33. [CY04b]
    C.-Q. Cheng and J. Yan. Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differential Geom., 67(3):457-517, 2004.MATHMathSciNetGoogle Scholar
  34. [DdlL06]
    D. Dolgopyat and R. de la Llave. Stochastic acceleration. Manusrcript, 2006.Google Scholar
  35. [DDLLS00]
    A. Delshams, R. de la Llave, and T. M. Seara. Unbounded growth of energy in periodic perturbations of geodesic flows of the torus. In Hamiltonian systems and celestial mechanics (Pátzcuaro, 1998), volume 6 of World Sci. Monogr. Ser. Math., pages 90-110. World Sci. Publ., River Edge, NJ, 2000.Google Scholar
  36. [DG96]
    A. Delshams and P. Gutiérrez. Effective stability and KAM theory. J. Differential Equations, 128(2):415-490, 1996.MATHMathSciNetGoogle Scholar
  37. [DH06]
    A. Delshams and G. Huguet. The large gap problem in arnold diffusion for non polynomial perturbations of an a-priori unstable hamiltonian system. Manuscript, 2006.Google Scholar
  38. [DLC83]
    R. Douady and P. Le Calvez. Exemple de point fixe elliptique non topologiquement stable en dimension 4. C. R. Acad. Sci. Paris Sér. I Math., 296(21):895-898, 1983.MATHMathSciNetGoogle Scholar
  39. [dlL06]
    R. de la Llave. Some recent progress in geometric methods in the instability problem in Hamiltonian mechanics. In International Congress of Mathematicians. Vol. II, pages 1705-1729. Eur. Math. Soc., Zürich, 2006.Google Scholar
  40. [dlLGJV05]
    R. de la Llave, A. González, À. Jorba, and J. Villanueva. KAM theory without action-angle variables. Nonlinearity, 18(2):855-895, 2005.MATHMathSciNetGoogle Scholar
  41. [dlLRR07]
    R. de la Llave and R. Ramirez-Ros. Instability in billiards with moving boundaries. Manuscript, 2007.Google Scholar
  42. [DLS00]
    A. Delshams, R. de la Llave, and T.M. Seara. A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T2. Comm. Math. Phys., 209(2):353-392, 2000.MATHMathSciNetGoogle Scholar
  43. [DLS03]
    Amadeu Delshams, Rafael de la Llave, and Tere M. Seara. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: announce-ment of results. Electron. Res. Announc. Amer. Math. Soc., 9:125-134 (electronic), 2003.MATHMathSciNetGoogle Scholar
  44. [DLS06a]
    A. Delshams, R. de la Llave, and T. M. Seara. Geometric properties of the scattering map to a normally hyperbolic manifold. Adv. Math., 2006. To appear.Google Scholar
  45. [DLS06b]
    A. Delshams, R. de la Llave, and T. M. Seara. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Mem. Amer. Math. Soc., 179(844):viii+141, 2006.MathSciNetGoogle Scholar
  46. [DLS06c]
    A. Delshams, R. de la Llave, and T. M. Seara. Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Adv. Math., 202(1):64-188, 2006.MATHMathSciNetGoogle Scholar
  47. [DLS07]
    A. Delshams, R. de la Llave, and T. M. Seara. Instability of high dimensional hamiltonian systems: multiple resonances do not impede diffusion. 2007.Google Scholar
  48. [Dou88]
    R. Douady. Regular dependence of invariant curves and Aubry-Mather sets of twist maps of an annulus. Ergodic Theory Dynam. Systems, 8(4):555-584, 1988.MATHMathSciNetGoogle Scholar
  49. [Dou89]
    R. Douady. Systèmes dynamiques non autonomes: démonstration d’un théorème de Pustyl’nikov. J. Math. Pures Appl. (9), 68(3):297-317, 1989.MATHMathSciNetGoogle Scholar
  50. [DR97]
    A. Delshams and R. Ramírez-Ros. Melnikov potential for exact symplectic maps. Comm. Math. Phys., 190:213-245, 1997.MATHMathSciNetGoogle Scholar
  51. [Eas78]
    R. W. Easton. Homoclinic phenomena in Hamiltonian systems with several degrees of freedom. J. Differential Equations, 29(2):241-252, 1978.MATHMathSciNetGoogle Scholar
  52. [Eas89]
    R. W. Easton. Isolating blocks and epsilon chains for maps. Phys. D, 39(1):95-110, 1989.MATHMathSciNetGoogle Scholar
  53. [EM79]
    R. W. Easton and R. McGehee. Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere. Indiana Univ. Math. J., 28(2):211-240, 1979.MATHMathSciNetGoogle Scholar
  54. [EMR01]
    R. W. Easton, J. D. Meiss, and G. Roberts. Drift by coupling to an anti-integrable limit. Phys. D, 156(3-4):201-218, 2001.MATHMathSciNetGoogle Scholar
  55. [Fen72]
    N. Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J., 21:193-226, 1971/1972.MATHMathSciNetGoogle Scholar
  56. [Fen77]
    N. Fenichel. Asymptotic stability with rate conditions. II. Indiana Univ. Math. J., 26(1):81-93, 1977.MATHMathSciNetGoogle Scholar
  57. [Fen79]
    N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations, 31(1):53-98, 1979.MATHMathSciNetGoogle Scholar
  58. [Fen74]
    N. Fenichel. Asymptotic stability with rate conditions. Indiana Univ. Math. J., 23:1109-1137, 1973/74.MathSciNetGoogle Scholar
  59. [FGL05]
    C. Froeschlé, M. Guzzo, and E. Lega. Local and global diffusion along resonant lines in discrete quasi-integrable dynamical systems. Celestial Mech. Dynam. Astronom., 92(1-3):243-255, 2005.MATHMathSciNetGoogle Scholar
  60. [FLG06]
    C. Froeschlé, E. Lega, and M. Guzzo. Analysis of the chaotic behaviour of orbits diffusing along the Arnold web. Celestial Mech. Dynam. Astronom., 95(1-4):141-153, 2006.MATHMathSciNetGoogle Scholar
  61. [FM00]
    E. Fontich and P. Martín. Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma. Nonlinearity, 13(5):1561-1593, 2000.MATHMathSciNetGoogle Scholar
  62. [FM01]
    E. Fontich and P. Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete Contin. Dynam. Systems, 7(1):61-84, 2001.MATHMathSciNetGoogle Scholar
  63. [FM03]
    E. Fontich and P. Martín. Hamiltonian systems with orbits covering densely submanifolds of small codimension. Nonlinear Anal., 52(1):315-327, 2003.MATHMathSciNetGoogle Scholar
  64. [FS07]
    R. Fontich, E. de la Llave and Y. Sire. Construction of invariant whiskered tori by a parameterization method. 2007. Manuscript.Google Scholar
  65. [Gar00]
    Antonio García. Transition tori near an elliptic fixed point. Discrete Contin. Dynam. Systems, 6(2):381-392, 2000.MATHMathSciNetGoogle Scholar
  66. [GL06a]
    M. Gidea and R. de la Llave. Arnold diffusion with optimal time in the large gap problem. Preprint, 2006.Google Scholar
  67. [GL06b]
    M. Gidea and R. de la Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete Contin. Dyn. Syst., 14(2):295-328, 2006.MATHMathSciNetGoogle Scholar
  68. [GLF05]
    M. Guzzo, E. Lega, and C. Froeschlé. First numerical evidence of global Arnold diffusion in quasi-integrable systems. Discrete Contin. Dyn. Syst. Ser. B, 5(3):687-698, 2005.MATHMathSciNetGoogle Scholar
  69. [GR04]
    M. Gidea and C. Robinson. Symbolic dynamics for transition tori II. In New advances in celestial mechanics and Hamiltonian systems, pages 95-109. Kluwer, Dordrecht, The Netherlands, 2004.Google Scholar
  70. [GZ04]
    M. Gidea and P. Zgliczy nski. Covering relations for multidimensional dynamical systems. II. J. Differential Equations, 202(1):59-80, 2004.MATHMathSciNetGoogle Scholar
  71. [Hal97]
    G. Haller. Universal homoclinic bifurcations and chaos near double resonances. J. Statist. Phys., 86(5-6):1011-1051, 1997.MATHMathSciNetGoogle Scholar
  72. [Hal99]
    G. Haller. Chaos near resonance. Springer, New York, 1999.MATHGoogle Scholar
  73. [HdlL06a]
    À. Haro and R. de la Llave. Manifolds on the verge of a hyperbolicity breakdown. Chaos, 16(1):013120, 8, 2006.MathSciNetGoogle Scholar
  74. [HdlL06b]
    À. 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(6):1261-1300 (electronic), 2006.MATHMathSciNetGoogle Scholar
  75. [HdlL06c]
    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(2):530-579, 2006.MATHMathSciNetGoogle Scholar
  76. [HdlL07]
    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(1):142-207 (electronic), 2007.MATHMathSciNetGoogle Scholar
  77. [Hed32]
    G. A. Hedlund. Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. of Math., 33:719-739, 1932.MathSciNetGoogle Scholar
  78. [Her83]
    M. R. Herman. Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 1, volume 103 of Astérisque. Société Mathématique de France, Paris, 1983.Google Scholar
  79. [HM82]
    P. J. Holmes and J. E. Marsden. Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys., 23(4):669-675, 1982.MATHMathSciNetGoogle Scholar
  80. [HP70]
    M. W. Hirsch and C. C. Pugh. Stable manifolds and hyperbolic sets. In S. Chern and S. Smale, editors, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pages 133-163, Amer. Math. Soc. Providence, RI, 1970.Google Scholar
  81. [HPS77]
    M. W. Hirsch, C. C. Pugh, and M. Shub. Invariant manifolds, volume 583 of Lecture Notes in Math. Springer, Berlin, 1977.Google Scholar
  82. [Hun68]
    W. Hunziker. The s-matrix in classical mechanics. Com. Math. Phys., 8(4):282-299, 1968.MATHMathSciNetGoogle Scholar
  83. [HW99]
    B. Hasselblatt and A. Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Ergodic Theory Dynam. Systems, 19(3):643-656, 1999.MATHMathSciNetGoogle Scholar
  84. [KP90]
    U. Kirchgraber and K. J. Palmer. Geometry in the neighborhood of invariant manifolds of maps and flows and linearization. Longman Scientific & Technical, Harlow, 1990.MATHGoogle Scholar
  85. [KPT95]
    T. Krüger, L. D. Pusty′nikov, and S. E. Troubetzkoy. Acceleration of bouncing balls in external fields. Nonlinearity, 8(3):397-410, 1995.MATHMathSciNetGoogle Scholar
  86. [Lev92]
    M. Levi. On Littlewood’s counterexample of unbounded motions in superquadratic potentials. In Dynamics reported: expositions in dynamical systems, pages 113-124. Springer, Berlin, 1992.Google Scholar
  87. [Lit66a]
    J. E. Littlewood. Unbounded solutions of an equation ÿ+ g(y) = p(t ), with p(t ) periodic and bounded, and g(y)/y → ∞ as y → ±∞. J. London Math. Soc., 41:497-507,1966.MATHMathSciNetGoogle Scholar
  88. [Lit66b]
    J. E. Littlewood. Unbounded solutions of ÿ + g(y) = p(t ). J. London Math. Soc., 41:491-496, 1966.MATHMathSciNetGoogle Scholar
  89. [LL91]
    S. Laederich and M. Levi. Invariant curves and time-dependent potentials. Ergodic Theory Dynam. Systems, 11(2):365-378, 1991.MATHMathSciNetGoogle Scholar
  90. [Lla01]
    R. de la Llave. A tutorial on KAM theory. In Smooth ergodic theory and its applications (Seattle, WA, 1999), pages 175-292. Amer. Math. Soc., Providence, RI, 2001.Google Scholar
  91. [Lla02]
    R. de la Llave. Orbits of unbounded energy in perturbations of geodesic flows by periodic potentials. a simple construction. Preprint, 2002.Google Scholar
  92. [Lla04]
    R. de la Llave. Orbits of unbounded energy in perturbation of geodesic flows: a simple mechanism. Preprint, 2004.Google Scholar
  93. [LM88]
    P. Lochak and C. Meunier. Multiphase Averaging for Classical Systems, volume 72 of Appl. Math. Sci. Springer, New York, 1988.Google Scholar
  94. [LMM86]
    R. de la Llave, J. M. Marco, and R. Moriyón. Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann. of Math. (2), 123(3): 537-611, 1986.MathSciNetGoogle Scholar
  95. [LMS03]
    P. Lochak, J.-P. Marco, and D. Sauzin. On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems. Mem. Amer. Math. Soc., 163(775):viii+145, 2003.MathSciNetGoogle Scholar
  96. [LP66]
    M. de La Place. Celestial mechanics. Vols. I-IV. Translated from the French, with a commentary, by Nathaniel Bowditch. Chelsea Publishing, Bronx, NY, 1966.Google Scholar
  97. [LT83]
    M. A. Lieberman and Jeffrey L. Tennyson. Chaotic motion along resonance layers in near-integrable Hamiltonian systems with three or more degrees of freedom. In C. Wendell Horton, Jr. and L. E. Reichl, editors, Long-time prediction in dynamics (Lakeway, Tex., 1981), pages 179-211. Wiley, New York, 1983.Google Scholar
  98. [LY97]
    M. Levi and J. You. Oscillatory escape in a Duffing equation with a polynomial potential. J. Differential Equations, 140(2):415-426, 1997.MATHMathSciNetGoogle Scholar
  99. [LZ95]
    M. Levi and E. Zehnder. Boundedness of solutions for quasiperiodic potentials. SIAM J. Math. Anal., 26(5):1233-1256, 1995.MATHMathSciNetGoogle Scholar
  100. [Mañ97]
    R. Mañé. Lagrangian flows: the dynamics of globally minimizing orbits. Bol. Soc. Brasil. Mat. (N.S.), 28(2):141-153, 1997.MATHMathSciNetGoogle Scholar
  101. [Mat93]
    J. N. Mather. Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble), 43(5):1349-1386, 1993.MATHMathSciNetGoogle Scholar
  102. [Mat96]
    J. N. Mather. Graduate course at Princeton, 95-96, and Lectures at Penn State, Spring 96, Paris, Summer 96, Austin, Fall 96.Google Scholar
  103. [Mey91]
    K. R. Meyer. Lie transform tutorial. II. In Kenneth R. Meyer and Dieter S. Schmidt, editors, Computer aided proofs in analysis (Cincinnati, OH, 1989), volume 28 of IMA Vol. Math. Appl., pages 190-210. Springer, New York, 1991.Google Scholar
  104. [Moe96]
    R. Moeckel. Transition tori in the five-body problem. J. Differential Equations, 129(2):290-314, 1996.MATHMathSciNetGoogle Scholar
  105. [Moe02]
    R. Moeckel. Generic drift on Cantor sets of annuli. In Celestial mechanics (Evanston, IL, 1999), volume 292 of Contemp. Math., pages 163-171. Amer. Math. Soc., Providence, RI, 2002.Google Scholar
  106. [Moe05]
    R. Moeckel. A variational proof of existence of transit orbits in the restricted three-body problem. Dyn. Syst., 20(1):45-58, 2005.MATHMathSciNetGoogle Scholar
  107. [Mor24]
    M. Morse. A fundamental class of geodesics on any closed surface of genus greater than one. Trans. Amer. Math. Soc., 26:26-60, 1924.MathSciNetGoogle Scholar
  108. [Mos69]
    J. Moser. On a theorem of Anosov. J. Differential Equations, 5:411-440, 1969.MATHMathSciNetGoogle Scholar
  109. [Mos73]
    J. Moser. Stable and random motions in dynamical systems. Princeton University Press, Princeton, NJ, 1973. With special emphasis on celestial mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, NJ, Annals Math. Studies, No. 77.Google Scholar
  110. [MS02]
    J.-P. Marco and D. Sauzin. Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems. Publ. Math. Inst. Hautes Études Sci., (96):199-275 (2003),2002.Google Scholar
  111. [Neı84]
    A. I. Neıshtadt. The separation of motions in systems with rapidly rotating phase. J. Appl. Math. Mech., 48(2):133-139, 1984.MathSciNetGoogle Scholar
  112. [Nel69]
    E. Nelson. Topics in dynamics. I: Flows. Mathematical Notes. Princeton University Press, Princeton, NJ, 1969.MATHGoogle Scholar
  113. [Nie07]
    L. Niederman. Prevalence of exponential stability among nearly integrable Hamiltonian systems. Ergodic Theory Dynam. Systems, 27(3):905-928, 2007.MATHMathSciNetGoogle Scholar
  114. [Ort97]
    R. Ortega. Nonexistence of invariant curves of mappings with small twist. Nonlinearity, 10(1):195-197, 1997.MATHMathSciNetGoogle Scholar
  115. [Ort04]
    R. Ortega. Unbounded motions in forced newtonian equations. Preprint, 2004.Google Scholar
  116. [Pal00]
    K. Palmer. Shadowing in dynamical systems, volume 501 of Mathematics and its Applications. Kluwer, Dordrecht, The Netherlands, 2000. Theory and applications.Google Scholar
  117. [Pes04]
    Y. B. Pesin. Lectures on partial hyperbolicity and stable ergodicity. Zurich Lectures in Advanced Mathematics. Eur. Math. Soc. (EMS), Zürich, 2004.Google Scholar
  118. [Pil99]
    S. Yu. Pilyugin. Shadowing in dynamical systems, volume 1706 of Lecture Notes in Mathematics. Springer, Berlin, 1999.Google Scholar
  119. [Poi99]
    H. Poincaré. Les méthodes nouvelles de la mécanique céleste, volume 1, 2, 3. Gauthier-Villars, Paris, 1892-1899.Google Scholar
  120. [Pol93]
    M. Pollicott. Lectures on ergodic theory and Pesin theory on compact manifolds, volume 180 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1993.Google Scholar
  121. [PS70]
    C. Pugh and M. Shub. Linearization of normally hyperbolic diffeomorphisms and flows. Invent. Math., 10:187-198, 1970.MATHMathSciNetGoogle Scholar
  122. [Pus77a]
    L. D. Pustyl’nikov. Stable and oscillating motions in nonautonomous dynamical systems. II. Trudy Moskov. Mat. Obš č ., 34:3-103, 1977.MATHMathSciNetGoogle Scholar
  123. [Pus77b]
    L. D. Pustyl’nikov. Stable and oscillating motions in nonautonomous dynamical systems. II. Trudy Moskov. Mat. Obš č ., 34:3-103, 1977. English translation: Trans. Moscow Math. Soc., 1978, Issue 2, pages 1-101. Amer. Math. Soc., Providence, RI, 1978,MATHMathSciNetGoogle Scholar
  124. [Pus95]
    L. D. Pustyl’nikov. Poincaré models, rigorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism. Uspekhi Mat. Nauk, 50(1(301)):143-186, 1995.MathSciNetGoogle Scholar
  125. [Rob71]
    C. Robinson. Differentiable conjugacy near compact invariant manifolds. Bol. Soc. Brasil. Mat., 2(1):33-44, 1971.MATHMathSciNetGoogle Scholar
  126. [Rob88]
    C. Robinson. Horseshoes for autonomous Hamiltonian systems using the Melnikov integral. Ergodic Theory Dynam. Systems, 8:395-409, 1988.Google Scholar
  127. [Rob02]
    C. Robinson. Symbolic dynamics for transition tori. In Celestial mechanics (Evanston, IL, 1999), volume 292 of Contemp. Math., pages 199-208. Amer. Math. Soc., Providence, RI, 2002Google Scholar
  128. [RS02]
    P. H. Rabinowitz and E. W. Stredulinsky. A variational shadowing method. In Celestial mechanics (Evanston, IL, 1999), volume 292 of Contemp. Math., pages 185-197. Amer. Math. Soc., Providence, RI, 2002Google Scholar
  129. [Sac65]
    R. J. Sacker. A new approach to the perturbation theory of invariant surfaces. Comm. Pure Appl. Math., 18:717-732, 1965.MATHMathSciNetGoogle Scholar
  130. [Shu78]
    M. Shub. Stabilité globale des systèmes dynamiques. Société Mathématique de France, Paris, 1978. With an English preface and summary.Google Scholar
  131. [Sim99]
    C. Simó, editor. Hamiltonian systems with three or more degrees of freedom, Kluwer, Dordrecht, The Netherlands, 1999Google Scholar
  132. [Sit53]
    K. A. Sitnikov. On the possibility of capture in the problem of three bodies. Mat. Sbornik N.S., 32(74):693-705, 1953.MathSciNetGoogle Scholar
  133. [Ten82]
    J. Tennyson. Resonance transport in near-integrable systems with many degrees of freedom. Phys. D, 5(1):123-135, 1982.MathSciNetGoogle Scholar
  134. [Thi83]
    W. Thirring. Classical scattering theory. In Conference on differential geometric methods in theoretical physics (Trieste, 1981), pages 41-64. World Sci. Publ., Singapore, 1983.Google Scholar
  135. [TLL80]
    J. L. Tennyson, M. A. Lieberman, and A. J. Lichtenberg. Diffusion in near-integrable Hamiltonian systems with three degrees of freedom. In Melvin Month and John C. Herrera, editors, Nonlinear dynamics and the beam-beam interaction (Sympos., Brookhaven Nat. Lab., New York, 1979), pages 272-301. Amer. Inst. Physics, New York, 1980.Google Scholar
  136. [Tre02a]
    D. Treschev. Multidimensional symplectic separatrix maps. J. Nonlinear Sci., 12(1):27-58, 2002.MATHMathSciNetGoogle Scholar
  137. [Tre02b]
    D. Treschev. Trajectories in a neighbourhood of asymptotic surfaces of a priori unstable Hamiltonian systems. Nonlinearity, 15(6):2033-2052, 2002.MATHMathSciNetGoogle Scholar
  138. [Tre04]
    D. Treschev. Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity, 17(5):1803-1841, 2004.MATHMathSciNetGoogle Scholar
  139. [Wei73]
    A. Weinstein. Lagrangian submanifolds and Hamiltonian systems. Ann. of Math. (2),98:377-410, 1973.MathSciNetGoogle Scholar
  140. [Wei79]
    A. Weinstein. Lectures on symplectic manifolds, volume 29 of CBMS Regional Conference Series in Mathematics. Amer. Math. Soc. Providence, RI, 1979. Corrected reprint.Google Scholar
  141. [Zas02]
    G. M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Phys. Rep., 371(6):461-580, 2002.MATHMathSciNetGoogle Scholar
  142. [ZG04]
    P. Zgliczy nski and M. Gidea. Covering relations for multidimensional dynamical systems. J. Differential Equations, 202(1):32-58, 2004.MathSciNetGoogle Scholar
  143. [ZZN+ 89]
    G. M. Zaslavskiı, M. Yu. Zakharov, A. I. Neıshtadt, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov. Multidimensional Hamiltonian chaos. Zh. Èksper. Teoret. Fiz., 96(11):1563-1586, 1989.Google Scholar

Copyright information

© Springer Science + Business Media B.V 2008

Authors and Affiliations

  • Amadeu Delshams
    • 1
  • Marian Gidea
    • 2
  • Rafael de la Llave
    • 3
  • Tere M. Seara
    • 1
  1. 1.Departament de Matemàtica AplicadaUniversitat Politècnica de CatalunyaSpain
  2. 2.Department of MathematicsNortheastern Illinois UniversityChicagoUSA
  3. 3.Department of MathematicsUniversity of TexasAustinUSA

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