Communications in Mathematical Physics

, Volume 348, Issue 1, pp 321–361 | Cite as

A Second Order Expansion of the Separatrix Map for Trigonometric Perturbations of a Priori Unstable Systems



In this paper we study a so-called separatrix map introduced by Zaslavskii–Filonenko (Sov Phys JETP 27:851–857, 1968) and studied by Treschev (Physica D 116(1–2):21–43, 1998; J Nonlinear Sci 12(1):27–58, 2002), Piftankin (Nonlinearity (19):2617–2644, 2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3–108, 2007). We derive a second order expansion of this map for trigonometric perturbations. In Castejon et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint available at arXiv:1501.03319, 2015), Guardia and Kaloshin (Stochastic diffusive behavior through big gaps in a priori unstable systems (in preparation), 2015), and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior for the generalized Arnold example away from resonances. Preprint available at, 2015), applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arn64.
    Arnold V.I.: Instability of dynamical systems with several degrees of freedom. Sov. Math. Dokl. 5, 581–585 (1964)Google Scholar
  2. BdlLW96.
    Banyaga A., de la Llave R., Wayne C.E.: Cohomology equations near hyperbolic points and geometric versions of Sternberg Linearization Theorem. J. Geom. Anal. 6(4), 613–649 (1996)MathSciNetMATHCrossRefGoogle Scholar
  3. Ber08.
    Bernard P.: The dynamics of pseudographs in convex Hamiltonian systems. J. Am. Math. Soc. 21(3), 615–669 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. BK05.
    Bourgain J., Kaloshin V.: Diffusion for Hamiltonian perturbations of integrable systems in high dimensions. J. Funct. Anal. 229, 1–61 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. BKZ11.
    Bernard, P., Kaloshin, V., Zhang, K.: Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders (2011). Preprint available at arXiv:1112.2773
  6. BT99.
    Bolotin S., Treschev D.: Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity 12(2), 365–388 (1999)ADSMathSciNetMATHCrossRefGoogle Scholar
  7. Che13.
    Cheng, C.Q.: Arnold diffusion in nearly integrable Hamiltonian systems (2013). Preprint available at arXiv:1207.4016
  8. Chi79.
    Chirikov B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52(5), 264–379 (1979)ADSMathSciNetCrossRefGoogle Scholar
  9. CK15.
    Castejon, O., Guardia, M., Kaloshin, V.: Random iteration of maps of a cylinder and diffusive behavior (2015). Preprint available at arXiv:1501.03319
  10. CY04.
    Cheng C.Q., Yan J.: Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differ. Geom. 67(3), 457–517 (2004)MathSciNetMATHGoogle Scholar
  11. CY09.
    Cheng C.Q., Yan J.: Arnold diffusion in hamiltonian systems: a priori unstable case. J. Differ. Geom. 82, 229–277 (2009)MathSciNetMATHGoogle Scholar
  12. CZ13.
    Cheng, C.Q., Zhang, J.: Asymptotic trajectories of KAM torus (2013). Preprint available at arXiv:1312.2102
  13. DdlLS00.
    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 \({\mathbb{T}^{2}}\). Commun. Math. Phys. 209(2), 353–392 (2000)ADSMathSciNetMATHCrossRefGoogle Scholar
  14. DdlLS06.
    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) (2006). doi: 10.1090/memo/0844
  15. DdlLS08.
    Delshams A., de la Llave R., Seara T.M.: Geometric properties of the scattering map of a normally hyperbolic invariant manifold. Adv. Math. 217(3), 1096–1153 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. DdlLS13.
    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)Google Scholar
  17. DG00.
    Delshams A., Gutiérrez P.: Splitting potential and the Poincaré–Melnikov method for whiskered tori in Hamiltonian systems. J. Nonlinear Sci. 10(4), 433–476 (2000)ADSMathSciNetMATHCrossRefGoogle Scholar
  18. DH09.
    Delshams A., Huguet G.: Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems. Nonlinearity 22(8), 1997–2077 (2009)ADSMathSciNetMATHCrossRefGoogle Scholar
  19. Eth05.
    Ethier, S.N., Kurtz, T.G.: Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)Google Scholar
  20. FGKP11.
    Fejoz, J., Guardia, M., Kaloshin, V., Roldan, P.: Kirkwood gaps and diffusion along mean motion resonance for the restricted planar three body problem (2011). Preprint available at arXiv:1109.2892
  21. GdlL06.
    Gidea M., de la Llave R.: Topological methods in the instability problem of Hamiltonian systems. Discrete Contin. Dyn. Syst. 14(2), 295–328 (2006)MathSciNetMATHGoogle Scholar
  22. GK14.
    Guardia, M., Kaloshin, V.: Orbits of nearly integrable systems accumulating to KAM tori (2014). Preprint available at arXiv:1412.7088
  23. GK15.
    Guardia, M., Kaloshin, V.: Stochastic diffusive behavior through big gaps in a priori unstable systems (2015) (in preparation)Google Scholar
  24. GT08.
    Gelfreich V., Turaev D.: Unbounded energy growth in Hamiltonian systems with a slowly varying parameter. Commun. Math. Phys. 283(3), 769–794 (2008)ADSMathSciNetMATHCrossRefGoogle Scholar
  25. Kal03.
    Kaloshin, V.: Geometric proofs of Mather’s accelerating and connecting theorems. In: London Mathematical Society, Lecture Notes Series, pp. 81–106. Cambridge University Press, Cambridge (2003)Google Scholar
  26. KL08a.
    Kaloshin V., Levi M.: An example of Arnold diffusion for near-integrable Hamiltonians. Bull. Am. Math. Soc. (N.S.) 45(3), 409–427 (2008)MathSciNetMATHCrossRefGoogle Scholar
  27. KL08b.
    Kaloshin V., Kaloshin V.: Geometry of Arnold diffusion. SIAM Rev. 50(4), 702–720 (2008)ADSMathSciNetMATHCrossRefGoogle Scholar
  28. KLS14.
    Kaloshin V., Levi M., Saprykina M.: Arnold diffusion in a pendulum lattice. Commun. Pure Appl. Math. 67(5), 748–775 (2014)MathSciNetMATHCrossRefGoogle Scholar
  29. KMV04.
    Kaloshin V., Mather J., Valdinoci E.: Instability of totally elliptic points of symplectic maps in dimension 4. Astérisque 297, 79–116 (2004)MathSciNetMATHGoogle Scholar
  30. KS12.
    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)ADSMathSciNetMATHCrossRefGoogle Scholar
  31. KZ12.
    Kaloshin, V., Zhang, K.: A strong form of Arnold diffusion for two and a half degrees of freedom (2012). Preprint available at
  32. KZZ15.
    Kaloshin, V., Zhang, J., Zhang, K.: Normally hyperbolic invariant laminations and diffusive behavior for the generalized Arnold example away from resonances (2015). Preprint available at
  33. Mat91a.
    Mather J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207(2), 169–207 (1991)MathSciNetMATHCrossRefGoogle Scholar
  34. Mat91b.
    Mather J.N.: Variational construction of orbits of twist diffeomorphisms. J. Am. Math. Soc. 4(2), 207–263 (1991)MathSciNetMATHCrossRefGoogle Scholar
  35. Mat93.
    Mather J.N.: Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble) 43(5), 1349–1386 (1993)MathSciNetMATHCrossRefGoogle Scholar
  36. Mat96.
    Mather, J.N.: Manuscript (1996) (unpublished)Google Scholar
  37. Mat03.
    Mather, J.N.: Arnold diffusion. I. Announcement of results. Sovrem. Mat. Fundam. Napravl. 2, 116–130 (2003) (electronic)Google Scholar
  38. Mat08.
    Mather, J.N.: Arnold diffusion II, 185 pp (2008) (preprint)Google Scholar
  39. Moe96.
    Moeckel R.: Transition tori in the five-body problem. J. Differ. Equ. 129(2), 290–314 (1996)ADSMathSciNetMATHCrossRefGoogle Scholar
  40. Mos56.
    Moser J.: The analytic invariants of an area-preserving mapping near a hyperbolic fixed point. Commun. Pure Appl. Math. 9, 673–692 (1956)MathSciNetMATHCrossRefGoogle Scholar
  41. MS02.
    Marco, J.P., Sauzin, D.: Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems. Publ. Math. Inst. Hautes Études Sci. 96, 199–275 (2003)Google Scholar
  42. MS04.
    Marco J.P., Sauzin D.: Wandering domains and random walks in Gevrey near integrable systems. Ergod. Theory Dyn. Syst. 24(5), 1619–1666 (2004)MathSciNetMATHCrossRefGoogle Scholar
  43. Pif06.
    Piftankin, G.: Diffusion speed in the Mather problem. Nonlinearity 19, 2617–2644 (2006)Google Scholar
  44. PT07.
    Piftankin G.N., Treshchëv D.V.: Separatrix maps in Hamiltonian systems. Uspekhi Mat. Nauk 62(2(374)), 3–108 (2007)MathSciNetMATHCrossRefGoogle Scholar
  45. Sau06.
    Sauzin, D.: Exemples de diffusion d’Arnold avec convergence vers un mouvement brownien (2006) (preprint)Google Scholar
  46. Sil65.
    Šil’nikov L.P.: A case of the existence of a denumerable set of periodic motions. Dokl. Akad. Nauk SSSR 160, 558–561 (1965)MathSciNetGoogle Scholar
  47. Tre98.
    Treschev D.: Width of stochastic layers in near-integrable two-dimensional symplectic maps. Physica D 116(1–2), 21–43 (1998)ADSMathSciNetMATHCrossRefGoogle Scholar
  48. Tre02.
    Treschev D.: Multidimensional symplectic separatrix maps. J. Nonlinear Sci. 12(1), 27–58 (2002)ADSMathSciNetMATHCrossRefGoogle Scholar
  49. Tre04.
    Treschev D.: Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17(5), 1803–1841 (2004)ADSMathSciNetMATHCrossRefGoogle Scholar
  50. Tre12.
    Treschev D.: Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems. Nonlinearity 9, 2717–2757 (2012)ADSMathSciNetMATHCrossRefGoogle Scholar
  51. ZF68.
    Zaslavskii G.M., Filonenko N.N.: Stochastic instability of trapped particles and conditions of applicability of the quasi-linear approximation. Sov. Phys. JETP 27, 851–857 (1968)ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.University of Maryland at College ParkCollege ParkUSA
  3. 3.University of TorontoTorontoCanada

Personalised recommendations