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Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter

  • Vassili GelfreichEmail author
  • Dmitry Turaev
Article

Abstract

We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows and Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from linear to exponential in time. Our theory also applies to non-Hamiltonian systems with a first integral.

Keywords

Periodic Orbit Hamiltonian System Invariant Manifold Heteroclinic Cycle Energy Growth 
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.

References

  1. 1.
    Afraimovich V.S., Shilnikov L.P.: On critical sets of Morse-Smale systems. Trans. Moscow Math. Soc. 28, 179–212 (1973)Google Scholar
  2. 2.
    Anosov, D.V.: Geodesic flows on closed Riemanian manifolds of negative curvature. Proc. Steklov Math. Inst. 90 (1967)Google Scholar
  3. 3.
    Arnold V.I.: Mathematical methods of classical mechanics. Springer-Verlag, New York (1989)Google Scholar
  4. 4.
    Bolotin S., Treschev D.: Unbounded growth of energy in nonautonomous Hamiltonian systems. Nonlinearity 12, 365–388 (1999)zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Bunimovich L.A., Sinai Ya.G., Chernov N.I.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45, 105–152 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    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 generic geodesic flows on \({\mathbb T^2}\). Commun. Math. Phys. 209, 353–392 (2000)zbMATHADSGoogle Scholar
  7. 7.
    Delshams A., de la Llave R., Seara T.M.: Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Adv. Math. 202, 64–188 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fenichel N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kaloshin V.: Geometric proofs of Mather’s connecting and accelerating theorems. London Math. Soc. Lecture Note Ser. 310, 81–106 (2003)MathSciNetGoogle Scholar
  10. 10.
    Kasuga T.: On the adiabatic theorem for the Hamiltonian system of differential equations in the classical mechanics. I, II, III. Proc. Japan Acad. 37, 366–382 (1961)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Katok, A., Strelcin, J.M.: Invariant manifolds, entropy and billiards—Smooth maps with singularities, Lecture Notes in Mathematics 1222, New York: Springer-Verlag, 1980Google Scholar
  12. 12.
    Lebovitz N.R., Neishtadt A.: Slow evolution in perturbed Hamiltonian systems. Stud. Appl. Math. 92, 127–144 (1994)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Loskutov A., Ryabov A.B., Akinshin L.G.: Properties of some chaotic billiards with time-dependent boundaries. J. Phys. A 33, 7973–7986 (2000)zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Loskutov A., Ryabov A.: Particle dynamics in time-dependent stadium-like billiards. J. Stat. Phys. 108, 995–1014 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Oliffson-Kamphorst S., Leonel E.D., da Silva J.K.L.: The presence and lack of Fermi acceleration in nonintegrable billiards. J. Phys. A: Math. Theor. 40, F887–F893 (2007)CrossRefADSGoogle Scholar
  16. 16.
    Piftankin G.N.: Diffusion speed in the Mather problem. Nonlinearity 19, 2617–2644 (2006)zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Rapoport A., Rom-Kedar V., Turaev D.: Approximating multi-dimensional Hamiltonian flows by billiards. Commun. Math. Phys. 272, 567–600 (2007)zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Sinai Ya.G.: Dynamical systems with elastic reflections: Ergodic properties of scattering billiards. Russ. Math. Sur. 25, 137–189 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sinai Ya.G., Chernov N.I.: Ergodic properties of some systems of two-dimensional disks and three-dimensional balls. Russ. Math. Sur. 42, 181–207 (1987)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Shilnikov L.P.: On a Poincaré-Birkhoff problem. Math. USSR Sb. 3, 91–102 (1967)CrossRefGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O.: Methods of qualitative theory in nonlinear dynamics. Part I. World Scientific, Singapore (1998)zbMATHGoogle Scholar
  23. 23.
    Treschev D.: Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17, 1803–1841 (2004)zbMATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Turaev D., Rom-Kedar V.: Islands appearing in near-ergodic flows. Nonlinearity 11, 575–600 (1998)zbMATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Brännström, N., Gelfreich, V.: Drift of slow variables in slow-fast Hamiltonian systems. Physica D (in press) (2008)Google Scholar
  26. 26.
    Gelfreich, V., Turaev, D.: Fermi acceleration in non-autonomous billiards. J. Phys. A: Math. Theor. 41, 212003 (6pp) (2008)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom
  2. 2.Department of MathematicsBen Gurion University of the NegevBe’er ShevaIsrael

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