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Foundations of Computational Mathematics

Erschienen in:

01.02.2015

Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values

verfasst von: Narcís Miguel, Carles Simó, Arturo Vieiro

Erschienen in: Foundations of Computational Mathematics | Ausgabe 1/2015

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Abstract

In this paper we review, based on massive, long-term, numerical simulations, the effect of islands on the statistical properties of the standard map for large parameter values. Different sources of discrepancy with respect to typical diffusion are identified, and their individual roles are compared and explained in terms of available limit models.

Vorheriger Artikel Overdetermined Systems of Sparse Polynomial Equations

Nächster Artikel A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points

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B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52 (1979), 264–379.

B- V. Chirikov, Chaotic dynamics in Hamiltonian systems with divided phase space, in Proceed. Sitges Conference on Dynamical Systems and Chaos (L. Garrido, ed.) Lecture Notes in Physics 179, Springer, 1983.

B. V. Chirikov and D. L. Shepelyansky, Statistics of Poincaré recurrences and the structure of the stochastic layer of a nonlinear resonance, Ninth international conference on nonlinear oscillations, Vol. 2 Kiev, 1981. Translation to English: Plasma physics laboratory, Princeton University, 1983.

B. V. Chirikov and D. L. Shepelyansky, Correlation properties of dynamical chaos in Hamiltonian systems, Physica 13 D (1984), 395–400.

G. Contopoulos and M. Harsoula, Stickiness effects in conservative systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2005–2043.

P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincar Anal. Non Linaire 11 (1994), 359–409.

F. Dumortier, S. Ibáñez, H. Kokubu and C. Simó, About the unfolding of a Hopf-zero singularity, Discrete Contin. Dyn. Syst. Ser. A 33 (2013), 4435–4471.

K. M. Frahm and D. L. Shepelyansky, Ulam method for the Chirikov standard map, Eur. Phys. J. B. 76 (2010), 57–68.

K. M. Frahm and D. L. Shepelyansky, Poincaré recurrences and Ulam method for the Chirikov standard map. Eur. Phys. J. B. 86 (2013), 322–333.

10.

A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem. J. Diff. Eq. 77 (1989), 167–198.

11.

M. Gonchenko, Homoclinic phenomena in conservative systems, Ph.D. Thesis, Universitat Politècnica de Catalunya, 2013.

12.

S. V. Gonchenko and L. P. Shilnikov, On two-dimensional area-preserving mappings with homoclinic tangencies, Russian Math. Dokl. 63 (2001), 395–399.

13.

J. M. Greene, A method for determining stochastic transition, J. Math. Phys. 6 (1979), 1183–1201.

14.

M. Hénon, Numerical study of quadratic area-preserving mappings, Quart. Appl. Math. 27 (1969), 291–312.

15.

I. Jungreis, A method for proving that monotone twist maps have no invariant circles, Ergod. Th. & Dynam. Sys. 11 (1991), 79–84.

16.

C. F. F. Karney, Long time correlations in the stochastic regime, Physica 3 D (1983), 360–380.

17.

C. F. F. Karney, A. Rechester and B. White, Effect of noise on the standard mapping, Physica 4 D (1982), 425–438.

18.

F. Ledrappier, M. Shub, C. Simó and A. Wilkinson, Random versus deterministic exponents in a rich family of diffeomorphisms, J. Stat Phys. 113 (2003), 85–149.

19.

A. J. Lichtenberg and M. A. Lieberman, Regular And Chaotic Dynamics, Applied Mathematical Sciences, 2nd edition, Springer, New York, 1992.

20.

R. S. MacKay, A renormalisation approach to invariant circles in area-preserving maps, Physica 7 D (1983), 283–300. Order in chaos (Los Alamos, N.M., 1982).

21.

R. S. MacKay, Renormalisation in area-preserving maps, Advanced Series in Nonlinear Dynamics, 6. World Scientific. 1992.

22.

R. S. MacKay, J. D. Meiss and I. C. Percival, Transport in Hamiltonian systems, Physica 13 D (1984), 55–81.

23.

R. S. MacKay and I. C. Percival, Converse KAM: theory and practice, Comm. Math. Phys. 98 (1985), 469–512.

24.

R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity 5 (1992), 867–888.

25.

S. Marmi and J. Stark, On the standard map critical function, Nonlinearity 5 (1992), 743–761.

26.

J. N. Mather, Nonexistence of invariant circles, Ergod. Th. & Dynam. Sys. 4 (1984), 301–309.

27.

J. D. Meiss, Class renormalization: Islands around islands, Phys. Rev. A 34 (1986), 2375–2383.

28.

J. D. Meiss, Average exit time for volume-preserving maps, Chaos 7 (1997), 139–147.

29.

J. D. Meiss, J. R. Cary, C. Grebogi, J. D. Crawford, A. N. Kaufman and H. D. I. Abarbanel, Correlations of Periodic Area-Preserving Maps, Physica 6 D (1983), 375–384.

30.

J. D. Meiss and E. Ott, Markov tree model of transport in area-preserving maps, Physica 20 D (1986), 387–402.

31.

N. Miguel, C. Simó, and A. Vieiro, From the Hénon conservative map to the Chirikov standard map for large parameter values, Regular and Chaotic Dynamics 20 (2013), 469–489.

32.

N. N. Nekhorosev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Mathematical Surveys 32 (1977), 1–65.

33.

A. Olvera and C. Simó, An obstruction method for the destruction of invariant curves, Physica 26 D (1987), 181–192.

34.

Ya. Pesin, Characteristic exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55–114.

35.

V. Rom-Kedar and G. Zaslavsky, Islands of accelerator modes and homoclinic tangles, Chaos 9 (1999), 697–705.

36.

J. Sánchez, M. Net and C. Simó, Computation of invariant tori by Newton-Krylov methods in large-scale dissipative systems, Physica 239 D (2010), 123–133.

37.

C. Simó, Analytical and numerical computation of invariant manifolds, in Modern methods in celestial mechanics (D. Benest and C. Froeschlé, eds), Editions Frontières, Paris, 1990, pp. 285–330.

38.

C. Simó, 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 Notes Series 363, 2009, New York, Cambridge University Press, pp. 163–189.

39.

C. Simó, P. Sousa-Silva and M. Terra, Practical Stability Domains near \(L_{4,5}\) in the Restricted Three-Body Problem: Some preliminary facts, in Progress and Challenges in Dynamical Systems, Vol. 54, Springer, 2013, pp. 367–382.

40.

C. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of Area Preserving Maps, Nonlinearity 22 (2009), 1191–1245.

41.

C. Simó and A. Vieiro, Dynamics in chaotic zones of area preserving maps: close to separatrix and global instability zones, Physica 240 D (2011), 732–753.

42.

R. Venegeroles, Calculation of superdiffusion for the Chirikov-Taylor model, Physical Review Letters 101 (2008): 054102.

43.

R. Venegeroles, Universality of Algebraic Laws in Hamiltonian Systems, Physical Review Letters 102 (2009): 064101.

44.

G. M. Zaslavsky, Dynamical traps, Physica 168–169 D (2002), 292–304.

45.

G. M. Zaslavsky, M. Edelman and B. A. Niyazov, Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics, Chaos 7 (1997), 159–181.

Titel: Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values
verfasst von: Narcís Miguel
Carles Simó
Arturo Vieiro
Publikationsdatum: 01.02.2015
Verlag: Springer US
Erschienen in: Foundations of Computational Mathematics / Ausgabe 1/2015
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-014-9210-3

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