Regular and Chaotic Dynamics

, Volume 19, Issue 6, pp 681–693 | Cite as

Hyperbolic sets near homoclinic loops to a saddle for systems with a first integral



A complete description of dynamics in a neighborhood of a finite bunch of homoclinic loops to a saddle equilibrium state of a Hamiltonian system is given.


Hamiltonian system nonintegrability and chaos resonance crossing Arnold diffusion 

MSC2010 numbers

37J30 37J40 37D05 37C29 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Turaev, D.V. and Shil’nikov, L.P., Hamiltonian Systems with Homoclinic Saddle Curves, Soviet Math. Dokl., 1989, vol. 39, no. 1, pp. 165–168; see also: Dokl. Akad. Nauk SSSR, 1989, vol. 304, no. 4, pp. 811–814.MATHMathSciNetGoogle Scholar
  2. 2.
    Shilnikov, L.P., On a Poincaré-Birkhoff Problem, Math. USSR-Sb., 1967, vol. 3, no. 3, pp. 353–371; see also: Mat. Sb. (N. S.), 1967, vol. 74(116), no. 3, pp. 378–397.CrossRefGoogle Scholar
  3. 3.
    Devaney, R. L., Homoclinic Orbits in Hamiltonian Systems, J. Differential Equations, 1976, vol. 21, no. 2, pp. 431–438.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Belyakov, L.A. and Shilnikov, L.P., Homoclinic Curves and Complex Solitary Waves, Selecta Math. Soviet., 1990, vol. 9, no. 3, pp. 219–228.MathSciNetGoogle Scholar
  5. 5.
    Lerman, L.M., Complex Dynamics and Bifurcations in a Hamiltonian System Having a Transversal Homoclinic Orbit to a Saddle Focus, Chaos, 1991, vol. 1, no. 2, pp. 174–180.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Buffoni, B. and Séré, E., A Global Condition for Quasi-Random Behavior in a Class of Conservative Systems, Comm. Pure Appl. Math., 1996, vol. 49, no. 3, pp. 285–305.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gelfreich, V., and Simó, C., and Vieiro, A., Dynamics of 4D Symplectic Maps near a Double Resonance, Phys. D, 2013, vol. 243, pp. 92–110.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Kaloshin, V. and Zhang, K., Normally Hyperbolic Invariant Manifolds near Strong Double Resonance, arXiv:1202.1032 (2012).Google Scholar
  9. 9.
    Kaloshin, V. and Zhang, K., A Strong form of Arnold Diffusion for Two and a Half Degrees of Freedom, arXiv:1212.1150 (2013).Google Scholar
  10. 10.
    Kaloshin, V. and Zhang, K., Partial Averaging and Dynamics of the Dominant Hamiltonian, with Applications to Arnold Diffusion, arXiv:1410-1844 (2014).Google Scholar
  11. 11.
    Bolotin, S.V. and Rabinowitz, P.H., A Variational Construction of Chaotic Trajectories for a Reversible Hamiltonian System, J. Differential Equations, 1998, vol. 148, no. 2, pp. 364–387.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Baesens, C., Chen, Y.-Ch., and MacKay, R. S., Abrupt Bifurcations in Chaotic Scattering: View from the Anti-Integrable Limit, Nonlinearity, 2013, vol. 26, no. 9, pp. 2703–2730.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Turaev, D., One Case of Bifurcations of a Contour Composed by Two Homoclinic Curves of a Saddle, in Methods of Qualitative Theory of Differential Equations, E.A. Leontovich (Ed.), Gorky: GGU, 1984, pp. 45–58 (Russian).Google Scholar
  14. 14.
    Sandstede, B., Center Manifolds for Homoclinic Solutions, J. Dynam. Differential Equations, 2000, vol. 12, no. 3, pp. 449–510.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Homburg, A. J., Global Aspects of Homoclinic Bifurcations of Vector Fields, Mem. Amer. Math. Soc., 1996, vol. 121, no. 578, 128 pp.Google Scholar
  16. 16.
    Turaev, D., On Dimension of Non-Local Bifurcational Problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1996, vol. 6, no. 5, pp. 919–948.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Shashkov, M. and Turaev, D., An Existence Theorem for a Nonlocal Invariant Manifold near a Homoclinic Loop, J. Nonlinear Sci., 1999, vol. 9, no. 5, pp. 1–49.CrossRefMathSciNetGoogle Scholar
  18. 18.
    Shilnikov, L.P., Shilnikov, A. L., Turaev, D., and Chua, L.O., Methods of Qualitative Theory in Nonlinear Dynamics: Part 1, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 4, River Edge, N.J.: World Sci., 1998.CrossRefMATHGoogle Scholar
  19. 19.
    Palis, J., On Morse-Smale Dynamical Systems, Topology, 1968, vol. 8, pp. 385–404.CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Imperial CollegeLondonUK
  2. 2.Lobachevsky State University of Nizhny NovgorodNizhny NovgorodRussia

Personalised recommendations