Regular and Chaotic Dynamics

, Volume 18, Issue 5, pp 469–489 | Cite as

From the Hénon conservative map to the Chirikov standard map for large parameter values



In this paper we consider conservative quadratic Hénon maps and Chirikov’s standard map, and relate them in some sense.

First, we present a study of some dynamical properties of orientation-preserving and orientation-reversing quadratic Hénon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects.

Then the phase space of the standard map, for large values of the parameter k, is studied. There are some stable orbits which appear periodically in k and are scaled somehow. Using this scaling, we show that the dynamics around these stable orbits is one of the above Hénon maps plus some small error, which tends to vanish as k→∞. Elementary considerations about diffusion properties of the standard map are also presented.


Hénon maps measure of regular and chaotic dynamics domains islands in the standard map for large parameter accelerator modes 

MSC2010 numbers

37J10 37J25 37J45 


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  1. 1.
    Newhouse, S.E., Diffeomorphisms with Infinitely Many Sinks, Topology, 1974, vol. 13, pp. 9–18.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Palis, J. and Takens, F., Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Stud. Adv. Math., vol. 35, Cambridge: Cambridge Univ. Press, 1995.Google Scholar
  3. 3.
    Gonchenko, S. V. and Shilnikov, L.P., On Two-Dimensional Area-Preserving Mappings with Homoclinic Tangencies, Dokl. Akad. Nauk, 2001, vol. 63, no. 3, pp. 395–399.Google Scholar
  4. 4.
    Gonchenko, S. V. and Shilnikov, L.P., On Two-Dimensional Area-Preserving Maps with Homoclinic Tangencies That Have Infinitely Many Generic Elliptic Periodic Points, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 2003, vol. 300, Teor. Predst. Din. Sist. Spets. Vyp. 8, pp. 155–166, 288–289 [J. Math. Sci. (N. Y.), 2005, vol. 128, no. 2, pp. 2767–2773].Google Scholar
  5. 5.
    Gonchenko, S.V. and Gonchenko, V. S., On Bifurcations of Birth of Closed Invariant Curves in the Case of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies, in Dynamical Systems and Related Problems of Geometry: Collected Papers Dedicated to the Memory of Academician A.A. Bolibrukh, Tr. Mat. Inst. Steklova, vol. 244, Moscow: Nauka, 2004, pp. 87–114 [Proc. Steklov Inst. Math., 2004, vol. 244, pp. 80–105].Google Scholar
  6. 6.
    Gonchenko, M., Homoclinic phenomena in conservative systems, Ph.D. Thesis, Universitat Politècnica de Catalunya, 2013.Google Scholar
  7. 7.
    Delshams, A., Gonchenko, S. V., Gonchenko, V. S., Lázaro, J. T., and Sten’kin, O., Abundance of Attracting, Repelling and Elliptic Periodic Orbits in Two-Dimensional Reversible Maps, Nonlinearity, 2013, vol. 26, pp. 1–33.CrossRefMATHGoogle Scholar
  8. 8.
    Chirikov, B.V., A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep., 1979, vol. 52, no. 5, pp. 264–379.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Karney, C. F. F., Rechester, A., and White, B., Effect of Noise on the Standard Mapping, Phys. D, 1982, vol. 4, no. 3, pp. 425–438.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hénon, M., Numerical Study of Quadratic Area-Preserving Mappings, Quart. Appl. Math., 1969, vol. 27, pp. 291–312.MathSciNetMATHGoogle Scholar
  11. 11.
    Simó, C. and Vieiro, A., Resonant Zones, Inner and Outer Splittings in Generic and Low Order Resonances of Area Preserving Maps, Nonlinearity, 2009, vol. 22, pp. 1191–1245.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Sánchez, J., Net, M., and Simó, C., Computation of Invariant Tori by Newton-Krylov Methods in Large-Scale Dissipative Systems, Phys. D, 2010, vol. 239, nos. 3–4, pp. 123–133.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Simó, C. and Vieiro, A., A Numerical Exploration of Weakly Dissipative Two-Dimensional Maps, in Proc. of ENOC (Eindhoven, Netherlands, 2005).Google Scholar
  14. 14.
    Simó, C., Some Properties of the Global Behaviour of Conservative Low Dimensional Systems, in Foundations of Computational Mathematics (Hong Kong, 2008), F. Cucker et al. (Eds.), London Math. Soc. Lecture Note Ser., vol. 363, Cambridge: Cambridge Univ. Press, 2009, pp. 163–189.Google Scholar
  15. 15.
    Dumortier, F., Ibáñez, S., Kokubu, H., and Simó, C., About the Unfolding of a Hopf-Zero Singularity, Discrete Contin. Dyn. Syst., 2013, vol. 33, no. 10, pp. 4435–4471.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fontich, E. and Simó, C., Invariant Manifolds for Near Identity Differentiable Maps and Splitting of Separatrices, Ergodic Theory Dynam. Systems, 1990, vol. 10, no. 2, pp. 319–346.MathSciNetMATHGoogle Scholar
  17. 17.
    Fontich, E. and Simó, C., The Splitting of Separatrices for Analytic Diffeomorphisms, Ergodic Theory Dynam. Systems, 1990, vol. 10, no. 2, pp. 295–318.MathSciNetMATHGoogle Scholar
  18. 18.
    Simó, C., Analytic and Numeric Computations of Exponentially Small Phenomena, in Proc. EQUADIFF (Berlin, 1999), B. Fiedler, K. Grögeri, and J. Sprekels (Eds.), Singapore: World Sci., 2000, pp. 967–976.Google Scholar
  19. 19.
    Gelfreich, V. and Simó, C., High-Precision Computations of Divergent Asymptotic Series and Homoclinic Phenomena, Discrete Contin. Dyn. Syst. Ser. B, 2008, vol. 10, nos. 2–3, pp. 511–536.MathSciNetMATHGoogle Scholar
  20. 20.
    Batut, C., Belabas, K., Bernardi, D., Cohen, H., and Olivier, M., Users’ Guide to PARI/GP,
  21. 21.
    Arnold, V. I. and Avez, A., Problèmes ergodiques de la mécanique classique, Paris: Gauthier-Villars, 1967.Google Scholar
  22. 22.
    Olvera, A. and Simó, C., An Obstruction Method for the Destruction of Invariant Curves, Phys. D, 1987, vol. 26, nos. 1–3, pp. 181–192.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Simó, C. and Treschev, D., Evolution of the “Last” Invariant Curve in a Family of Area Preserving Maps, Preprint, 1998; see also Google Scholar
  24. 24.
    Simó, C., Invariant Curves of Perturbations of Non Twist Integrable Area Preserving Maps, Regul. Chaotic Dyn., 1998, vol. 3, no. 3, pp. 180–195.MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Simó, C. and Vieiro, A., Dynamics in Chaotic Zones of Area Preserving Maps: Close to Separatrix and Global Instability Zones, Phys. D, 2011, vol. 240, no. 8, pp. 732–753.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Chirikov, B.V. and Izraelev, F.M., Some Numerical Experiments with a Nonlinear Mapping: Stochastic Component, in Colloques internationaux du C.N.R. S. transformations ponctuelles et leurs applications, Toulouse, 1973.Google Scholar
  27. 27.
    Simó, C., Analytical and Numerical Computation of Invariant Manifolds, in Modern Methods in Celestial Mechanics, D. Benest, C. Froeschlé (Eds.), Ed. Frontières, 1990, pp. 285–330.Google Scholar
  28. 28.
    Simó, C. and Treschev, D., Stability Islands in the Vicinity of Separatrices of Near-Integrable Symplectic Maps, Discrete Contin. Dyn. Syst. Ser. B, 2008, vol. 10, nos. 2–3, pp. 681–698.MathSciNetMATHGoogle Scholar
  29. 29.
    Simó, C. and Vieiro, A., Some Remarks on the Abundance of Stable Periodic Orbits Inside Homoclinic Lobes, Phys. D, 2011, vol. 240, pp. 1936–1953.MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Greene, J. M., A Method for Determining Stochastic Transition, J. Math. Phys., 1979, vol. 20, no. 6, pp. 1183–1201.CrossRefGoogle Scholar
  31. 31.
    Zaslavsky, G. M., Zakharov, M.Yu., Sagdeev, R. Z., Usikov, D.A., and Chernikov, A.A., Stochastic Web and Diffusion of Particles in Magnetic Field, Zh. Eksp. i Teor. Fiz., 1986, vol. 91, pp. 500–516 [Sov. Phys. JETP, 1986, vol. 64, pp. 294–303].MathSciNetGoogle Scholar
  32. 32.
    Rom-Kedar, V. and Zaslavsky, G., Islands of Accelerator Modes and Homoclinic Tangles, Chaos, 1999, vol. 9, no. 3, pp. 697–705.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Chirikov, B.V., Chaotic Dynamics in Hamiltonian Systems with Divided Phase Space, in Proc. Sitges Conf. on Dynamical Systems and Chaos, L. Garrido (Ed.), Lect. Notes Phys. Monogr., vol. 179, Berlin: Springer, 1983.Google Scholar
  34. 34.
    Lichtenberg, A. J. and Lieberman, M.A., Regular and Chaotic Dynamics, 2nd ed., Appl. Math. Sci., vol. 38, New York: Springer, 1992.CrossRefMATHGoogle Scholar
  35. 35.
    Miguel, N., Simó, C., and Vieiro, A., On the Effect of Islands in the Diffusive Properties of the Standard Map, for Large Parameter Values, submitted to publication, 2013.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelona, CatalunyaSpain

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