Abstract
This report is a continuation of an analysis, initiated elsewhere V.V. Vecheslavov and B. V. Chirikov, Zh. Éksp. Teor. Fiz. 114, 1516 (1998) [JETP 86, 823 (1998)], of the effect of splitting of the separatrix of a nonlinear resonance for the model of standard mapping, based on results of direct measurements of the splitting angle α(K), where K is the system parameter. Measurements were made in the previously used wide range 0.1≳α≳10−208 (1⩾K⩾0.0004), but with significantly higher relative (better than 1050) and average (∼10−55) accuracy. This procedure made it possible to substantially refine the effects observed in Ref. 1 and construct qualitatively new empirical dependences providing reliable extrapolation of the data obtained for the angle and the invariant in the intermediate asymptotic limit K≲10−2 beyond the limits of the investigated region. The results obtained by us can be useful for further development of the theory of separatrix splitting and formation of the stochastic layer of a nonlinear resonance.
Similar content being viewed by others
References
V. V. Vecheslavov and B. V. Chirikov, Zh. Éksp. Teor. Fiz. 114, 1516 (1998) [JETP JETP 86, 823 (1998)].
B. V. Chirikov, Phys. Rep. 52, 263 (1979).
G. M. Zaslavskii and B. V. Chirikov, Usp. Fiz. Nauk 105, 3 (1971) [Sov. Phys. Usp. 14, 549 (1972)].
R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: from the Pendulum to Turbulence and Chaos (Harwood, Chur, 1988).
A. Lichtenberg and M. Lieberman, Regular and Chaotic Dynamics (Springer, 1992).
A. Poincaré, Les méhodes nouvelles de la mécanique céleste (Paris, 1892), p. 226.
V. K. Mel’nikov, Trudy Mosk. Mat. Obshchestva 12, 3 (1963).
N. N. Filonenko, R. Z. Sagdeev, and G. M. Zaslavsky, Nucl. Fusion 7, 253 (1967).
G. M. Zaslavskii and N. N. Filonenko, Zh. Éksp. Teor. Fiz. 54, 1590 (1968) [Sov. Phys. JETP 27, 851 (1968)].
V. F. Lazutkin, “Splitting of the separatrix of the standard Chirikov mapping” [in Russian] Dep. VINITI 6372-84 (1984); V. F. Lazutkin, I. G. Schachmanski, and M. B. Tabanov, Physica D 40, 235 (1989); V. G. Gelfreich, V. F. Lazutkin, and M. B. Tabanov, Chaos 1, 137 (1991).
V. G. Gelfreich, V. F. Lazutkin, and N. V. Svanidze, Physica D 71, 82 (1994).
V. G. Gelfreich, “A proof of the exponentially small transversality of the separatrices for the standard map.” Freie Universität Berlin, Preprint 9/98 (1998), p. 56 (to appear in Commun. Math. Phys.).
D. H. Bailey, ACM Transactions on Mathematical Software, Vol. 19 (1993), p. 288.
V. G. Gelfreich, private communication.
B. M. Shchigolev, Mathematical Processing of Observations [in Russian] (Fizmatgiz, Moskva, 1960); D. Hudson, Statistics (Geneva, 1964)
C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, N. J., 1957).
Author information
Authors and Affiliations
Additional information
Zh. Éksp. Teor. Fiz. 116, 336–346 (July 1999)
Rights and permissions
About this article
Cite this article
Vecheslavov, V.V. Fine structure of splitting of the separatrix of a nonlinear resonance. J. Exp. Theor. Phys. 89, 182–187 (1999). https://doi.org/10.1134/1.558967
Received:
Issue Date:
DOI: https://doi.org/10.1134/1.558967