Splitting of separatrices for the Chirikov standard map

V. F. Lazutkin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems. Part VIII, Tome 300 (2003), pp. 25-55

Résumé

This paper is an English translation (made by V. Gelfreich) of V. F. Lazutkin's work that was published in 1984 by VINITI and thus was not easily available for readers. In the paper, a formula for an exponentially small angle of separatrix splitting of the Chirikov standard map was obtained for the first time.

Bibliographie

[1] A. N. Kolmogorov, “On conservation of conditionally periodic motions for a small change in Hamilton's function”, Dokl. Akad. Nauk SSSR, 98 (1954), 527–530 | MR | Zbl

[2] V. I. Arnol'd, “Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian”, Uspekhi Mat. Nauk, 18:5(113) (1963), 13–40 | MR

[3] J. Moser, “On invariant curves of area-preserving mappings of an annulus”, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962, 1–20 | MR | Zbl

[4] V. F. Lazutkin, “On Moser's invariant curves theorem”, Problems of Dynamical Theory of Seismic Waves, Leningrad, 1974, 109–120 | MR

[5] N. V. Svanidze, “Small perturbations of an integrable dynamical system with an integral invariant”, Boundary value problems of mathematical physics, Trudy Steklov Mat. Inst., 147, 1980, 124–146 | MR | Zbl

[6] J. Pöschel, “Integrability of Hamiltonian systems on Cantor sets”, Comm. Pure Appl. Math., 35:5 (1982), 653–696 | DOI | MR

[7] A. J. Lichtenberg, M. A. Lieberman, Regular and stochastic motion, Applied Mathematical Sciences, 38, Springer-Verlag, New York, 1983 | MR | Zbl

[8] J. M. Mather, “Existence of quasiperiodic orbits for twist homeomorphisms of the annulus”, Topology, 21:4 (1982), 457–467 | DOI | MR | Zbl

[9] A. Katok, “Periodic and quasiperiodic orbits for twist maps”, Dynamical systems and chaos (Sitges/Barcelona, 1982), Lecture Notes Phys., 179, Springer, Berlin, 1983, 47–65 | MR

[10] B. V. Chirikov, Studies on the theory of nonlinear resonances and stochasticity, Institute of Nuclear Physics, Preprint No 267, Novosibirsk, 1969

[11] B. V. Chirikov, Interaction of nonlinear resonances, Novosibirsk State University, Novosibirsk, 1978

[12] A. I. Neishtadt, “On the accuracy of conservation of the adiabatic invariant”, J. Appl. Math. Mech., 45:1 (1981), 80–87 | DOI | MR

[13] H. Poincaré, Selected works, V. 2, Moscow, 1972 | Zbl

[14] S. Aubry, “Many defect structure, stochasticity and incommensurability”, Physique des Défaults, Les Houches. Session XXXV (1980), North-Holland Publ. Company, 1981, 432–451

[15] A. Aubry, “The devil's staircase transformation in incommensurate lattices”, The Riemann problem, complete integrability and arithmetic applications (Bures-sur-Yvette, New York, 1979–1980), Lecture Notes Math., 925, Springer, Berlin–New York, 1982, 221–245 | MR

[16] S. Aubry, P. Y. Le Daeron, “The discrete Frenkel-Kontorova model and its extensions. I: Exact results for the ground-states”, Phys. D, 8:3 (1983), 381–422 | DOI | MR

[1] J. M. Greene, I. C. Percival, “Hamiltonian maps in the complex plane”, Physica D, 3 (1981), 530–548 | DOI | MR | Zbl

[2] V. F. Lazutkin, Splitting of separatrices for the Chirkov standard map, Preprint VINITI 6372-84, 1984 | MR

[3] V. F. Lazutkin, M. B. Tabanov, I. G. Schakhmanskii, Splitting of separatrices for the standard and semistandard mappings, Preprint VINITI 4027-85, 1985 | Zbl

[4] V. F. Lazutkin, “Splitting of separatrices for a standard family of area-preserving transformations”, Wave propagation. Scattering theory, Probl. Mat. Fiz., 12, Leningrad. Univ., Leningrad, 1987, 32–41 ; 256–257 | MR

[5] V. F. Lazutkin, “Splitting of complex separatrices”, Funkt. Anal. Prilozhen., 22:2 (1988), 83–84 | DOI | MR

[6] V. F. Lazutkin, “Analytic integrals of a semistandard mapping, and separatrix splitting”, Algebra Analiz, 1:2 (1989), 116–131 | MR

[7] V. F. Lazutkin, I. G. Schachmannski, M. B. Tabanov, “Splitting of separatrices for standard and semistandard mappings”, Physica D, 40:2 (1989), 235–248 | DOI | MR | Zbl

[8] V. F. Lazutkin, “The width of the instability zone around separatrices of a standard mapping”, Dokl. Akad. Nauk SSSR, 313:2 (1990), 268–272 | MR

[9] V. F. Lazutkin, “Analytic integrals and separatrices splittings in Hamiltonian systems”, Conference on Differential Equations and Applications (Ruse, 1989), Tech. Univ., Ruse, 1991, 116–123 | MR

[10] V. G. Gelfreich, V. F. Lazutkin, M. B. Tabanov, “Exponentially small splittings in Hamiltonian systems”, Chaos, 1:2 (1991), 137–142 | DOI | MR | Zbl

[11] V. F. Lazutkin, “An analytic integral along the separatrix of the semistandard map: existence and an exponential estimate for the distance between the stable and unstable separatrices”, Algebra Analiz, 4:4 (1992), 110–142 | MR

[12] V. F. Lazutkin, “Recent results on the separatrix splitting for the standard map”, Sémin. Théor. Spectr. Géom., 11, Univ. Grenoble I, Saint-Martin-d'Hres, 1993 | MR

[13] V. F. Lazutkin, “Resurgent approach to the separatrices splitting”, International Conference on Differential Equations, Vol. 1, 2 (Barcelona, 1991), World Sci. Publishing, River Edge, NJ, 1993, 163–176 | MR | Zbl

[14] V. G. Gelfreich, V. F. Lazutkin, N. V. Svanidze, “A refined formula for the separatrix splitting for the standard map”, Physica D, 71:1–2 (1994), 82–101 | DOI | MR | Zbl

[15] V. F. Lazutkin, Splitting of separatrices for the Chirikov's standard map, Mathematical Physics Preprint Archive 98-421, 1998 | MR

[16] V. G. Gelfreich, “A proof of the exponentially small transversality of the separatrices for the standard map”, Comm. Math. Phys., 201:1 (1999), 155–216 | DOI | MR | Zbl

[17] V. G. Gelfreich, V. F. Lazutkin, “Splitting of separatrices: perturbation theory and exponential smallness”, Uspekhi Mat. Nauk, 56:3(339) (2001), 79–142 | MR