Instability of resonant totally elliptic points of symplectic maps in dimension 4

Asterisque

Volumn , Issue 297, 2004, Pages 79-116

  Kaloshin, Vadim ^a,b   Mather, John N ^c   Valdinoci, Enrico ^d  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^c PRINCETON UNIVERSITY   (United States)

^d UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

Arnold diffusion; Lyapunov stability of periodic orbits; Mather theory; Minimal measures; Variational methods

Indexed keywords

EID: 30344461845     PISSN: 03031179     EISSN: None     Source Type: Book Series    
DOI: None     Document Type: Article

References (32)

1. [Ar1] V. ARNOLD - "On the Stability of Positions of Equilibrium of a Hamiltonian System of Ordinary Differential Equations in the General Elliptic Case", Dokl. Akad. Nauk SSSR 137 (1961), no. 2, p. 255-257,
2. Sov Math Dokl 2 (1961), p. 247-279.
3. [Ar2] _, Mathematical Methods in Classical Mechanics, Graduate Texts in Math., vol. 60, Springer-Verlag, 1989.
4. [AKN] V. ARNOLD, V. KOZLOV & A. NEIDSTADT - Mathematical aspects of classical and celestial mechanics, Encyclopedia Math. Sci., vol. 3, Springer, Berlin, 1993, Translated from the 1985 Russian original by A. Iacob.
5. [Be] P. BERNARD - "The dynamics of pseudographs in convex hamiltonian systems", preprint, 56 pp., http://www-fourier.ujf-grenoble.fr/~pbernard/textes/PG.pdf, 200-1.
6. [BK] D. BERNSTEIN & A. KATOK - "Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonian", Invent. Math. 88 (1987), p. 225-241.
7. [CY] CH-Q. CHENG & J. YAN - "Existence of Diffusion Orbits in a priori Unstable Hamiltonian systems", to appear in J. Differential Geometry, 53 pp.
8. [DLS] A. DELSHAMS, R. DE LA LLAVE & T. SEARA - "A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristic and rigorous verification on a model", Electron. Res. Announc. Amer. Math. Soc. 9 (2003), p. 125-134, electronic.
9. [DC] M. DIAS CARNEIRO - "On minimizing measures of the action of autonomous Lagrangians", Nonlinearity 8 (1995), p. 1077-1085.
10. [Dou] R. DOUADY - "Stabilité ou Instabilité des Points Fixes Elliptiques", Ann. scient. Éc. Norm. Sup. 4° série 21 (1988), p. 1-40.
11. [Fa] A. FATHI - "Weak KAM Theorem in Lagrangian Dynamics", preprint of a forthcoming book, 139 pp., October 2003.
12. [Go] CH. GOLÉ - Symplectic Twist Maps, Global Variational Techniques, Advanced Series in Nonlinear Dynamics, vol. 18, 2001.
13. [Hed] G.A. HEDLUND - "The dynamics of geodesic flows", Bull. Amer. Math. Soc. (N.S.) 45 (1939), p. 241-260.
14. [Her] M. HERMAN - "Dynamics connected to indefinite normal torsion", IMA, vol. 44, Springer-Verlag.
15. [KM] V. KALOSHIN & J. MATHER - "Instabilities of nearly integrable a priori unstable Hamiltonian systems", in preparation.
16. [KP] S. KUKSIN & J. PÖSCIIEL - "On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications", in Seminar on Dynamical Systems (St. Petersburg, 1991), Progr. Nonlinear Differential Equations Appl., vol. 12, Birkhäuser, Basel, 1994, p. 96-116.
17. [LM] M. LEVI & J. MOSER - "A Lagrangian proof of the invariant curve theorem for twist mappings", in Smooth ergodic theory and Us applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, American Mathematical Society, Providence, RI, 2001, p. 733-746.
18. [Ma] J. MATHER - "Action minimizing invariant measures for positive Lagrangian systems", Math. Z. 207 (1991), no. 2, p. 169-207.
19. [Ma2] _, "Variational construction of connecting orbits", Ann. Inst. Fourier (Grenoble) 43 (1993), p. 1349-1386.
20. [Ma3] _, "Existence of unbounded orbits for generic mechanical systems on 2-torus", preprint, 1996.
21. [Ma4] _, Graduate class at Princeton, 2002-2003.
22. [Ma5] _, "Arnold diffusion, I: Announcement of results", Kluwer Academic Plenum Public, ser. Journ. of Math. Sciences (2004).
23. [McS] D. MCDUFF & D. SALAMON - Introduction to Symplectic Topology, Oxford Mathematical Monographs, 1995.
24. [MH] K. MEYER & G. HALL - Introduction to Hamiltonian dynamical systems and the N-body problem, Springer-Verlag, New York, 1995.
25. [Mor] M. MORSE - "A fundamental class of geodesics on any closed surface of genus greater than one", Trans. Amer. Math. Soc. 26 (1924), p. 25-60.
26. [Mo] J. MOSER - "On Invariant curves of Area-Preserving Mappings of an Annulus", Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1 (1962).
27. [Mo2] _, "Monotone twist mappings and the calculus of variations", Ergodic Theory Dynam. Systems 6 (1986), no. 3, p. 401-413.
28. [T1] D. TRESCHEV - "Multidimensional symplectic separatrix maps", J. Nonlinear Sci. 12 (2002), no. 1, p. 27-58.
29. [T2] _, "Trajectories in a neighborhood of asymptotic surfaces of a priori unstable Hamiltonian systems", Nonlinearity 15 (2002), no. 6, p. 2033-2052.
30. [T3] _, "Evolution of slow variables in a priori unstable Hamiltonian systems", preprint, 34pp.
31. [X1] Z. XIA - "Arnold diffusion: a variational construction", in Proc. of ICM, vol. II, (Berlin, 1998), 1988.
32. [X2] _, "Arnold diffusion and instabilities in hamiltonian dynamics", preprint.