Separatrix maps in Hamiltonian systems

G. N. Piftankin   ; D. V. Treschev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 2, pp. 219-322

Résumé

The separatrix map is constructed for some classes of problems in Hamiltonian dynamics. The formulae obtained are used to study two-dimensional symplectic maps close to integrable maps: elliptic periodic trajectories passing through separatrix lobes are constructed, and some estimates for the width of the stochastic layer are given. For Hamiltonian systems with two and a half degrees of freedom it is proved that the Arnol'd diffusion in the a priori unstable case is generic, and in the Mather problem trajectories are constructed for which the mean energy growth is linear in time.

Bibliographie

[1] G. M. Zaslavskii, N. N. Filonenko, “Stokhasticheskaya neustoichivost zakhvachennykh chastits i usloviya primenimosti kvazilineinogo priblizheniya”, ZhETF, 54 (1968), 1590–1602

[2] L. P. Shilnikov, “Ob odnom sluchae suschestvovaniya schetnogo mnozhestva periodicheskikh dvizhenii”, Dokl. AN SSSR, 160:3 (1965), 558–561 | MR | Zbl

[3] J. M. Greene, “A method for determining a stochastic transition”, J. Math. Phys., 20:6 (1979), 1183–1201 | DOI

[4] R. S. MacKay, I. C. Percival, “Converse KAM: Theory and practice”, Comm. Math. Phys., 98:4 (1985), 469–512 | DOI | MR | Zbl

[5] A. Olvera, C. Simó, “An obstruction method for the destruction of invariant curves”, Phys. D, 26:1–3 (1987), 181–192 | DOI | MR | Zbl

[6] N. N. Filonenko, R. Z. Sagdeev, G. M. Zaslavsky, “Destruction of magnetic surfaces by magnetic field irregularities. II”, Nucl. Fusion, 7 (1967), 253–266

[7] B. V. Chirikov, “A universal instability of many-dimensional oscillation systems”, Phys. Rep., 52:5 (1979), 263–379 | DOI | MR

[8] S. A. Dovbysh, “Struktura Kolmogorovskogo mnozhestva vozle separatris ploskogo otobrazheniya”, Matem. zametki, 46:4 (1989), 112–114 | MR | Zbl

[9] T. Ahn, G. Kim, S. Kim, “Analysis of the separatrix map in Hamiltonian systems”, Phys. D, 89:3–4 (1996), 315–328 | DOI | MR | Zbl

[10] V. F. Lazutkin, “O shirine zony ustoichivosti vozle separatris standartnogo otobrazheniya”, Dokl. AN SSSR, 313:2 (1990), 268–272 | MR | Zbl

[11] Ya. G. Sinai, “Neskolko tochnykh rezultatov ob ubyvanii korrelyatsii”, Dopolnenie v kn. G. M. Zaslavskii, Statisticheskaya neobratimost v nelineinykh sistemakh, Sovrem. problemy fiziki, Nauka, M., 1970, 124–139

[12] V. V. Kozlov, “Rasscheplenie separatris i rozhdenie izolirovannykh periodicheskikh reshenii v gamiltonovykh sistemakh s polutora stepenyami svobody”, UMN, 41:5 (1986), 177–178 | MR | Zbl

[13] D. V. Treschev, “Hyperbolic tori and asymptotic surfaces in Hamiltonian systems”, Russian J. Math. Phys., 2:1 (1994), 93–110 | MR | Zbl

[14] S. A. Dovbysh, “Rasscheplenie separatris i rozhdenie izolirovannykh periodicheskikh reshenii v gamiltonovykh sistemakh s polutora stepenyami svobody”, UMN, 44:2 (1989), 229–230 | MR | Zbl

[15] C. Simó, D. Treschev, “Stability islands in the vicinity of separatrices of near-integrable symplectic maps”, J. Discrete Contin. Dyn. Syst. (to appear)

[16] V. I. Arnold, “O neustoichivosti dinamicheskikh sistem so mnogimi stepenyami svobody”, Dokl. AN SSSR, 156:1 (1964), 9–12 | MR | Zbl

[17] N. N. Nekhoroshev, “Eksponentsialnaya otsenka vremeni ustoichivosti gamiltonovykh sistem blizkikh k integriruemym”, UMN, 32:6(198) (1977), 5–66 | MR | Zbl

[18] R. Douady, “Stabilité ou instabilité des points fixes elliptiques”, Ann. Sci. École Norm. Sup. (4), 21:1 (1988), 1–46 | MR | Zbl

[19] J.-P. Marco, D. Sauzin, “Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems”, Publ. Math. Inst. Hautes Études Sci., 96:1 (2003), 199–275 | DOI | MR | Zbl

[20] E. Fontich, P. Martin, “Arnold diffusion in perturbations of analytic integrable Hamiltonian systems”, Discrete Contin. Dynam. Systems, 7:1 (2001), 61–84 | MR | Zbl

[21] G. Gallavotti, G. Gentile, V. Mastropietro, “Hamilton–Jacobi equation, heteroclinic chains and Arnol'd diffusion in three time scale systems”, Nonlinearity, 13:2 (2000), 323–340 | DOI | MR | Zbl

[22] C. Simó , C. Valls, “A formal approximation of the splitting of separatrices in the classical Arnold's example of diffusion with two equal parameters”, Nonlinearity, 14:6 (2001), 1707–1760 | DOI | MR | Zbl

[23] J. N. Mather, “Arnold duffision. I: Announcement of results”, J. Math. Sci. (N. Y.), 124:5 (2004), 5275–5289 | DOI | MR | Zbl

[24] G. N. Piftankin, “Diffusion speed in the Mather problem”, Nonlinearity, 19:11 (2006), 2617–2644 | DOI | MR | Zbl

[25] S. Bolotin, D. Treschev, “Unbounded growth of energy in nonautonomous Hamiltonian systems”, Nonlinearity, 12:2 (1999), 365–388 | DOI | MR | Zbl

[26] A. Delshams, R. de la Llave, T. M. Seara, “A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by potential of generic geodesic flows on $T^2$”, Comm. Math. Phys., 209:2 (2000), 353–392 | MR | Zbl

[27] V. Kaloshin, “Geometric proofs of Mather's connecting and accelerating theorems”, Topics in dynamics and ergodic theory (Katsiveli, Ukraine, 2000), London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, 2003, 81–106 | MR | Zbl

[28] L. Chierchia, G. Gallavotti, “Drift and diffusion in phase space”, Ann. Inst. H. Poincaré. Phys. Théor., 60:1 (1994), 1–144 | MR | Zbl

[29] D. Treschev, “Multidimensional symplectic separatrix maps”, J. Nonlinear Sci., 12:1 (2002), 27–58 | DOI | MR | Zbl

[30] D. Treschev, “Evolution of slow variables in a priori unstable Hamiltonian systems”, Nonlinearity, 17:5 (2004), 1803–1841 | DOI | MR | Zbl

[31] M. Berti, P. Bolle, “Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11:4 (2000), 235–243 | MR | Zbl

[32] J. Cresson, C. Guillet, “Periodic orbits and Arnold diffusion”, Discrete Contin. Dyn. Syst., 9:2 (2003), 451–470 | MR | Zbl

[33] M. Berti, L. Biasco, P. Bolle, “Drift in phase space: A new variational mechanism with optimal diffusion time”, J. Math. Pures Appl. (9), 82:6 (2003), 613–664 | DOI | MR | Zbl

[34] B. Berti, L. Biasco, P. Bolle, “Optimal stability and instability results for a class of nearly integrable Hamiltonian systems”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 13:2 (2002), 77–84 | MR | Zbl

[35] Z. Xia, “Arnold diffusion: a variational construction”, Doc. Math. J. DMV Extra Volume ICM II, 1998, 867–877 | MR | Zbl

[36] C.-Q. Cheng, J. Yan, “Existence of diffusion orbits in a priori unstable Hamiltonian systems”, J. Differential Geom., 67:3 (2004), 457–517 | MR | Zbl

[37] A. Delshams, R. de la Llave, T. M. Seara, A geometric mechanism for diffusion in Hamiltonian systems, http://www.ma.utexas.edu/mp_arc/e/03-182.ps

[38] A. Delshams, R. de la Llave, T. M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristic and rigorous verification on a model, Mem. Amer. Math. Soc., 179, no. 844, Amer. Math. Soc., Providence, RI, 2006 | MR | Zbl

[39] Dzh. D. Birkgof, Dinamicheskie sistemy, Gostekhizdat, M.–L., 1941 ; G. D. Birkhoff, Dynamical systems, Amer. Math. Soc., New York, 1927 | Zbl | MR | Zbl | Zbl

[40] J. Moser, “The analytic invariants of an area-preserving mapping near a hyperbolic fixed point”, Comm. Pure Appl. Math., 9:4 (1956), 673–692 | DOI | MR | Zbl

[41] A. Banyaga, R. de la Llave, C. E. Wayne, “Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem”, J. Geom. Anal., 6:4 (1996), 613–649 | MR | Zbl

[42] D. V. Treschev, Vvedenie v teoriyu vozmuschenii gamiltonovykh sistem, Fazis, M., 1998 | MR

[43] S. L. Ziglin, “Vetvlenie reshenii i nesuschestvovanie pervykh integralov v gamiltonovoi mekhanike. I, II”, Funkts. analiz i ego prilozh., 16:3 (1982), 30–41 ; 17:1 (1983), 8–23 ; S. L. Ziglin, “Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I, II”, Funct. Anal. Appl., 16:3 (1982), 181–189 ; 17:1 (1983), 6–17 | MR | Zbl | MR | Zbl | DOI | DOI

[44] A. Pronin, D. Treschev, “On the inclusion of analytic maps into analytic flows”, Regul. Chaotic Dyn., 2:2 (1997), 14–24 | MR | Zbl

[45] D. Treschev, “Width of stochastic layers in near-integrable two-dimensional symplectic maps”, Phys. D, 116:1–2 (1998), 21–43 | DOI | MR | Zbl

[46] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, “Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki”, Itogi nauki i tekhniki. Sovrem. problemy matem. Fundam. napr., 3, VINITI, M., 1985, 5–304 ; V. I. Arnol'd, V. V. Kozlov, A. I. Neĭshtadt, Mathematical aspects of classical and celestial mechanics, Dynamical systems, III, 3rd ed., Encyclopaedia Math. Sci., 3, Springer, Berlin, 2006 | MR | Zbl | MR | Zbl

[47] S. M. Graff, “On the conservation of hyperbolic tori for Hamiltonian systems”, J. Differential Equations, 15:1 (1974), 1–69 | DOI | MR | Zbl

[48] E. Zehnder, “Generalized implicit function theorem with applications to some small divisor problems. I, II”, Comm. Pure Appl. Math., 28:1 (1975), 91–140 ; 29:1 (1976), 49–111 | MR | Zbl | DOI | MR | Zbl

[49] H. Poincaré, Les métodes nouvelles de la mécanique céleste, 1–3, Gauthier-Villars, Paris, 1892 ; 1893 ; 1899 | Zbl | Zbl | Zbl

[50] A. Delshams, P. Gutiérrez, “Splitting potential and Poincaré–Melnikov method for whiskered tori in Hamiltonian systems”, J. Nonlinear Sci., 10:4 (2000), 433–476 | DOI | MR | Zbl

[51] V. G. Gelfreich, “Separatrices splitting for the rapidly forced pendulum”, Seminar on Dynamical Systems (St. Petersburg, Russia, 1991), Progr. Nonlinear Differential Equations Appl., 12, Birkhäuser, Basel, 1994, 47–67 | MR | Zbl

[52] V. F. Lazutkin, Rasscheplenie separatris v standartnom otobrazhenii Chirikova, dep. v VINITI No 6372-84

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

[54] V. Gelfreikh, V. Lazutkin, “Rasscheplenie separatris: teoriya vozmuschenii, eksponentsialna malost”, UMN, 56:3 (2001), 79–142 | MR | Zbl

[55] A. Delshams, T. M. Seara, “An asymptotic expressions for the splitting of separatrices of the rapidly forced pendulum”, Comm. Math. Phys., 150:3 (1992), 433–463 | DOI | MR | Zbl

[56] D. Treschev, “An averaging method for Hamiltonian systems, exponentially close to integrable ones”, Chaos, 6:1 (1996), 6–14 | DOI | MR | Zbl

[57] D. Treschev, “Separatrix splitting for a pendulum with rapidly oscillating suspension point”, Russian J. Math. Phys., 5:1 (1997), 63–98 | MR | Zbl

[58] S. Aubry, “The twist map, the extended Frenkel–Kontorova model and the devil's staircase”, Phys. D, 7:1–3 (1983), 240–258 | DOI | MR | Zbl

[59] R. S. MacKay, “A renormalization approach to invariant circles in area-preserving maps”, Phys. D, 7:1–3 (1983), 283–300 | DOI | MR | Zbl

[60] J. N. Mather, “Non-existence of invariant circles”, Ergodic Theory Dynam. Systems, 4:2 (1984), 301–311 | DOI | MR | Zbl

[61] V. G. Gelfreich, “Reference systems for splitting of separatrices”, Nonlinearity, 10:1 (1997), 175–193 | DOI | MR | Zbl

[62] G. D. Birkhoff, “Surface transformations and their dynamical applications”, Acta Math., 43 (1920), 1–119 | DOI | MR | Zbl

[63] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, vol. I, Astérisque, 103–104, Soc. Math. France, Paris, 1983 | MR | Zbl

[64] D. Treschev, “Trajectories in a neighborhood of asymptotic surface of a priori unstable Hamiltonian systems”, Nonlinearity, 15:6 (2002), 2033–2052 | DOI | MR | Zbl