Abstract
We develop canonical perturbation theory for a physically interesting class of infinite-dimensional systems. We prove stability up to exponentially large times for dynamical situations characterized by a finite number of frequencies. An application to two model problems is also made. For an arbitrarily large FPU-like system with alternate light and heavy masses we prove that the exchange of energy between the optical and the acoustical modes is frozen up to exponentially large times, provided the total energy is small enough. For an infinite chain of weakly coupled rotators we prove exponential stability for two kinds of initial data: (a) states with a finite number of excited rotators, and (b) states with the left part of the chain uniformly excited and the right part at rest.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Fröhlich, T. Spencer, and C. E. Wayne, Localization in disordered, nonlinear dynamical systems,J. Stat. Phys. 42:247–274 (1986).
M. Vittot and J. Bellissard, Invariant tori for an infinite lattice of coupled classical rotators, Preprint (1985).
J. Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems,Commun. Math. Phys. 127:351–393 (1990).
S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum,Funkt. Anal. Prilozhen. 21(3):22–37 (1987) [Funct. Anal. Appl. 21 (1987)].
C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equation via KAM theory,Commun. Math. Phys. 127:479–528 (1990).
L. Chierchia and P. Perfetti, Maximal almost periodic solutions for Lagrangian equations on infinite dimensional tori, Preprint (1992).
S. B. Kuksin, Perturbation theory for quasiperiodic solutions of infinite dimensional Hamiltonian systems. 1. Symplectic structures and Hamiltonian scales of Hilbert spaces, Max Planck Institut für Mathematik, preprint MPI/90-99; Perturbation theory for quasiperiodic solutions of infinite dimensional Hamiltonian systems. 2. Statement of the main theorem and its consequences, Max Planck Institut für Mathematik, preprint MPI/90-100.
G. Benettin, J. Fröhlich, and A. Giorgilli, A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom,Commun. Math. Phys. 119:95–108 (1988).
D. Bambusi, A Nekhoroshev-type theorem for the Pauli-Fierz model of classical electrodynamics, Department of Mathematics, University of Milan, preprint 43/1991.
S. B. Kuksin, An averaging theorem for distributed conservative systems and its application to Von Karman's equation,PMM USSR 53:150–157 (1989).
G. Benettin, L. Galgani, and A. Giorgilli, Classical perturbation theory for systems of weakly coupled rotators,Nuovo Cimento B 89:89–102 (1985).
G. Benettin, L. Galgani, and A. Giorgilli, Numerical investigations on a chain of weakly coupled rotators in the light of classical perturbation theory,Nuovo Cimento B 89:103–119 (1985).
C. E. Wayne, The KAM theory of systems with short range interaction, I,Commun. Math. Phys. 96:311–329 (1984); The KAM theory of systems with short range interaction, II,Commun. Math. Phys. 96:331–344 (1984).
C. E. Wayne, On the elimination of non-resonant harmonics,Commun. Math. Phys. 103:351–386 (1986); C. E. Wayne, Bounds on the trajectories of a system of weakly coupled rotators,Commun. Math. Phys. 104:21–36 (1986).
L. Galgani, A. Giorgilli, A. Martinoli, and S. Vanzini, On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: Analytical and numerical estimates,Physica D 59:334–348 (1992).
G. Benettin, L. Galgani, and A. Giorgilli, Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory, Part II,Commun. Math. Phys. 121:557–601 (1989).
P. Lochak, Canonical perturbation theory via simultaneous approximation,Uspekhi Math. Nauk, to appear.
M. P. Chernoff and J. E. Marsden,Properties of Infinite Dimensional Hamiltonian Systems (Springer-Verlag, Berlin, 1974).
Y. Choquet-Bruhat, C. De Witt-Morette, and M. Dillard-Bleick,Analysis, Manifolds and Physics (North-Holland, Amsterdam, 1978).
A. Giorgilli and L. Galgani, Formal integrals for an autonomous Hamiltonian system near an equilibrium point,Cel. Mech. 17:267–2800 (1978).
S. Lang,Differential Manifolds (Springer-Verlag, New York, 1985).
J. Mujica,Complex Analysis in Banach Spaces (North-Holland, Amsterdam, 1986).
J. Moser, Regularization of Kepler's problem and the averaging method on a manifold,Commun. Pure Appl. Math. 23:609–636 (1970).
R. Cushman, Normal form of Hamiltonian vectorfields with periodic flows, inDifferential Geometric Methods in Mathematical Physics, S. Sterneberg, ed. (Reidel, Dordrecht, 1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bambusi, D., Giorgilli, A. Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems. J Stat Phys 71, 569–606 (1993). https://doi.org/10.1007/BF01058438
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01058438