Diffusion times and stability exponents for nearly integrable analytic systems

• Autorzy: Pierre Lochak , Jean-Pierre Marco
• Języki publikacji: EN

Abstrakty

EN

For a positive integer n and R>0, we set $$B_R^n = \left\{ {x \in \mathbb{R}^n |\left\| x \right\|_\infty< R} \right\}$$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $$h\left( r \right) = \tfrac{1}{2}r_1^2 + ...\tfrac{1}{2}r_{n - 1}^2 + r_n $$ on $$\mathbb{T}^n \times B_R^n $$ , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of $$\mathbb{T}^n \times B_R^n $$ , and setting $$\varepsilon _j : = \left\| {h - H_j } \right\|_{C^0 (V)} $$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.

Słowa kluczowe

EN

Perturbations; normal forms; small divisors; Arnol'd diffusion; 37J40; 37B; 37D

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 3
• Strony: 342-397

Daty

wydano

2005-09-01

online

2005-09-01

Twórcy

autor

Pierre Lochak

• Université Paris VI, UMR 7586

autor

Jean-Pierre Marco

• Université Paris VI, UMR 7586

Bibliografia

• [1] V.I. Arnold: “Instability of dynamical systems with several degrees of freedom”, Dokl. Akad. Nauk SSSR, Vol. 156, (1964), pp. 9–12; Soviet Math. Dokl., Vol. 5, (1964), pp. 581–585.
• [2] A. Banyaga, R. de la Llave and C.E. Wayne: “Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem”, J. Geom. Anal., Vol. 6, (1996), pp. 613–649.
• [3] A. Banyaga, R. de la Llave and C.E. Wayne: “Cohomology equations and commutators of germs of contact diffeomorphisms”, Trans. AMS, Vol. 312, (1989), pp. 755–778. http://dx.doi.org/10.2307/2001010
• [4] G. Benettin and G. Gallavotti: “Stability of motions near resonances in quasiintegrable Hamiltonian systems”, J. Phys. Stat., Vol. 44, (1986), pp. 293–338. http://dx.doi.org/10.1007/BF01011301
• [5] G. Benettin, L. Galgani and A. Giorgilli: “A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems”, Celestial Mech., Vol. 37, (1985), pp. 1–25. http://dx.doi.org/10.1007/BF01230338
• [6] U. Bessi: “An approach to Arnold's diffusion through the calculus of variations”, Nonlinear Anal. TMA, Vol. 26, (1996), pp. 1115–1135. http://dx.doi.org/10.1016/0362-546X(94)00270-R
• [7] U. Bessi: “Arnold's example with three rotators”, Nonlinearity, Vol. 10, (1997), pp. 763–781. http://dx.doi.org/10.1088/0951-7715/10/3/010
• [8] J. Bourgain: “Diffusion for Hamiltonian perturbations of integrable systems in high dimensions”, (2003), preprint.
• [9] J. Bourgain and V. Kaloshin: “Diffusion for Hamiltonian perturbations of integrable systems in high dimensions”, (2004), preprint.
• [10] M. Chaperon: “Géométrie différentielle et singularités de systèmes dynamiques”, Astérisque, Vol. 138–139, (1986).
• [11] M. Chaperon: “Invariant manifolds revisited”, Tr. mat. inst. Steklova, Vol. 236, (2002), Differ. Uravn. i. Din. Sist., pp. 428–446.
• [12] M. Chaperon: “Stable manifolds and the Perron-Irwin method”, Erg. Th. and Dyn. Syst., Vol. 24, (2004), pp. 1359–1394. http://dx.doi.org/10.1017/S0143385703000701
• [13] M. Chaperon and F. Coudray: “Invariant manifolds, conjugacies and blow-up”, Erg. Th. and Dyn. Syst., Vol. 17, (1997), pp. 783–791. http://dx.doi.org/10.1017/S0143385797085052
• [14] B.V. Chirikov: “A universal instability of many-dimensional oscillator systems”, Phys. Reports, Vol. 52, (1979), pp. 263–379. http://dx.doi.org/10.1016/0370-1573(79)90023-1
• [15] R. Douady: “Stabilité ou instabilité des points fixes elliptiques”, Ann. Sc. Éc. Norm. Sup., Vol. 21, (1988), pp. 1–46.
• [16] R. Easton and R. McGehee: “Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere”, Ind. Univ. Math. Journ., Vol. 28(2), (1979).
• [17] E. Fontich and P. Mart′ in: “Arnold diffusion in perturbations of analytic integrable Hamiltonian systems”, Discrete and Continuous Dyn. Syst., Vol. 7(1), (2001), pp. 61–84.
• [18] G. Gallavotti: “Quasi-integrable mechanical systems”, In: K. Osterwalder and R. Stora (Eds.): Phénomènes criliques, systèmes aléatoires, théories de jauge, part II (Les Houches 1984), North-Holland, Amsterdam New York, 1986, pp. 539–624.
• [19] M. Herman: Notes de travail, December, 1999, manuscript.
• [20] M.W. Hirsch, C.C. Pugh and M. Shub: Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer Verlag, 1977.
• [21] S. Kuksin and J. Pöschel: “Nekhoroshev estimates for quasi-convex Hamiltonian systems”, Math. Z., Vol. 213, (1993), pp. 187–216.
• [22] P. Lochak: “Canonical perturbation theory via simultaneous approximation”, Usp. Mat. Nauk., Vol. 47 (1992), pp. 59–140; Russian Math. Surveys, Vol. 47, (1992), pp. 57–133.
• [23] P. Lochak, J.-P. Marco and D. Sauzin: “On the splitting of invariant manifolds in multi-dimensional near-integrable Hamiltonian systems” Memoirs of the Amer. Math. Soc., Vol. 163, (2003).
• [24] P. Lochak and A.I. Neishtadt: “Estimates in the theorem of N. N. Nekhorocheff for systems with a quasi-convex Hamiltonian”, Chaos, Vol. 2, (1992), pp. 495–499. http://dx.doi.org/10.1063/1.165891
• [25] P. Lochak, A.I. Neishtadt and L. Niederman: “Stability of nearly integrable convex Hamiltonian systems over exponentially long times”, In: Proc. 1991 Euler Institute Conf. on Dynamical Systems, Birkhaüser, Boston, 1993.
• [26] J.-P. Marco: “Transition le long des chaines de tores invariants pour les systèmes hamiltoniens analytiques”, Ann. Inst. H. Poincaré, Vol. 64(2), (1996), pp. 205–252.
• [27] J.-P. Marco: “Uniform lower bounds of the splitting for analytic symplectic systems”, Ann. Inst. Fourier, submitted to.
• [28] J.-P. Marco, and D. Sauzin: “Stability and instability for Gevrey quasi-convex nearintegrable Hamiltonian systems”, Publ. Math. I.H.E.S., Vol. 96, (2003), pp. 77.
• [29] N.N. Nekhoroshev: “An exponential estimate of the time of stability of nearly integrable Hamiltonian systems”, Usp. Mal. Nauk., Vol. 32, (1977), pp. 5–66; Russian Math. Surveys, Vol. 32, (1977), pp. 1–65.
• [30] J. Pöschel: “Nekhoroshev estimates for quasi-convex Hamiltonian systems”, Math. Z., Vol. 213, (1993), pp. 187–216. http://dx.doi.org/10.1007/BF03025718
• [31] S. Sternberg: “Local contractions and a theorem of Poincaré”, Amer. J. Math., Vol. 79, (1957), pp. 809–824. http://dx.doi.org/10.2307/2372437
• [32] S. Sternberg: “On the structure of local homeomorphisms II”, Amer. J. Math., Vol. 80, (1958), pp. 623–631. http://dx.doi.org/10.2307/2372774
• [33] S. Sternberg: “On the structure of local homeomorphisms III”, Amer. J. Math., Vol. 81, (1959), pp. 578–604. http://dx.doi.org/10.2307/2372915
• [34] S. Sternberg: “Infinite Lie groups and the formal aspects of dynamical systems”, J. Math. Mech., Vol. 10, (1961), pp. 451–474.
• [35] D. V. Treshchev: “Evolution of slow variables in near-integrable Hamiltonian systems”, Progress in nonlinear science, Vol. 1, (2001), pp. 166–169.

Identyfikatory

DOI

10.2478/BF02475913