×

From the Hénon conservative map to the Chirikov standard map for large parameter values. (English) Zbl 1417.37191

Summary: 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 \(k\to\infty\). Elementary considerations about diffusion properties of the standard map are also presented.

MSC:

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37C05 Dynamical systems involving smooth mappings and diffeomorphisms

Software:

PARI/GP
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Newhouse, S.E., Diffeomorphisms with Infinitely Many Sinks, Topology, 1974, vol. 13, pp. 9–18. · Zbl 0275.58016
[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. · Zbl 0790.58014
[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. · Zbl 1041.37033
[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].
[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]. · Zbl 1079.37046
[6] Gonchenko, M., Homoclinic phenomena in conservative systems, Ph.D. Thesis, Universitat Politècnica de Catalunya, 2013.
[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. · Zbl 1277.37044
[8] Chirikov, B.V., A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep., 1979, vol. 52, no. 5, pp. 264–379.
[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. · Zbl 1194.60050
[10] Hénon, M., Numerical Study of Quadratic Area-Preserving Mappings, Quart. Appl. Math., 1969, vol. 27, pp. 291–312. · Zbl 0191.45403
[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. · Zbl 1181.37077
[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. · Zbl 1183.37137
[13] Simó, C. and Vieiro, A., A Numerical Exploration of Weakly Dissipative Two-Dimensional Maps, in Proc. of ENOC (Eindhoven, Netherlands, 2005).
[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.
[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. · Zbl 1283.34039
[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. · Zbl 0706.58060
[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. · Zbl 0706.58061
[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. · Zbl 0963.65136
[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. · Zbl 1169.37013
[20] Batut, C., Belabas, K., Bernardi, D., Cohen, H., and Olivier, M., Users’ Guide to PARI/GP, http://pari.math.u-bordeaux.fr/
[21] Arnold, V. I. and Avez, A., Problèmes ergodiques de la mécanique classique, Paris: Gauthier-Villars, 1967.
[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. · Zbl 0612.58039
[23] Simó, C. and Treschev, D., Evolution of the ”Last” Invariant Curve in a Family of Area Preserving Maps, Preprint, 1998; see also http://www.maia.ub.es/dsg/1998/index.html
[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. · Zbl 0932.37048
[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. · Zbl 1217.37051
[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.
[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.
[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. · Zbl 1153.37406
[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. · Zbl 1244.37039
[30] Greene, J. M., A Method for Determining Stochastic Transition, J. Math. Phys., 1979, vol. 20, no. 6, pp. 1183–1201.
[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].
[32] Rom-Kedar, V. and Zaslavsky, G., Islands of Accelerator Modes and Homoclinic Tangles, Chaos, 1999, vol. 9, no. 3, pp. 697–705. · Zbl 0987.37049
[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. · Zbl 0532.70011
[34] Lichtenberg, A. J. and Lieberman, M.A., Regular and Chaotic Dynamics, 2nd ed., Appl. Math. Sci., vol. 38, New York: Springer, 1992. · Zbl 0748.70001
[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. · Zbl 1376.37017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.