132 research outputs found
KAM Tori for 1D Nonlinear Wave Equations with Periodic Boundary Conditions
In this paper, one-dimensional (1D) nonlinear wave equations , with periodic boundary conditions are considered; V is a
periodic smooth or analytic function and the nonlinearity f is an analytic
function vanishing together with its derivative at u=0. It is proved that for
``most'' potentials V(x), the above equation admits small-amplitude periodic or
quasi-periodic solutions corresponding to finite dimensional invariant tori for
an associated infinite dimensional dynamical system. The proof is based on an
infinite dimensional KAM theorem which allows for multiple normal frequencies.Comment: 30 page
Explicit estimates on the measure of primary KAM tori
From KAM Theory it follows that the measure of phase points which do not lie
on Diophantine, Lagrangian, "primary" tori in a nearly--integrable,
real--analytic Hamiltonian system is , if
is the size of the perturbation. In this paper we discuss how the constant in
front of depends on the unperturbed system and in
particular on the phase--space domain
The Steep Nekhoroshev's Theorem
Revising Nekhoroshev's geometry of resonances, we provide a fully
constructive and quantitative proof of Nekhoroshev's theorem for steep
Hamiltonian systems proving, in particular, that the exponential stability
exponent can be taken to be ) ('s
being Nekhoroshev's steepness indices and the number of degrees of
freedom)
The spin-orbit resonances of the Solar system: A mathematical treatment matching physical data
In the mathematical framework of a restricted, slightly dissipative
spin-orbit model, we prove the existence of periodic orbits for astronomical
parameter values corresponding to all satellites of the Solar system observed
in exact spin-orbit resonance
On the measure of KAM tori in two degrees of freedom
A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small
measure of the non-torus set in analytic systems with two degrees of freedom is
discussed
Global properties of generic real-analytic nearly-integrable Hamiltonian systems
We introduce a new class of generic real analytic potentials
on and study global analytic properties of natural
nearly-integrable Hamiltonians , with potential
, on the phase space
with a given ball in . The phase space can be
covered by three sets: a `non-resonant' set, which is filled up to an
exponentially small set of measure (where is the maximal size of
resonances considered) by primary maximal KAM tori; a `simply resonant set' of
measure and a third set of measure
which is `non perturbative', in the sense that the -dynamics on it can be
described by a natural system which is {\sl not} nearly-integrable. We then
focus on the simply resonant set -- the dynamics of which is particularly
interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori)
-- and show that on such a set the secular (averaged) 1 degree-of-freedom
Hamiltonians (labelled by the resonance index ) can be put
into a universal form (which we call `Generic Standard Form'), whose main
analytic properties are controlled by {\sl only one parameter, which is uniform
in the resonance label }
Isolated Diophantine numbers
In this short note, we discuss the topology of Diophantine numbers, giving
simple explicit examples of Diophantine isolated numbers (among those with same
Diophantine constatnts), showing that, Diophantine sets are not always Cantor
sets. General properties of isolated Diophantine numbers are also briefly
discussed
Analytic Lagrangian tori for the planetary many-body problem
In 2004, F\'ejoz [D\'emonstration du 'th\'eor\'eme d'Arnold' sur la
stabilit\'e du syst\`eme plan\'etaire (d'apr\`es M. Herman). Ergod. Th. &
Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's
[D\'emonstration d'un th\'eor\'eme de V.I. Arnold. S\'eminaire de Syst\'emes
Dynamiques et manuscripts, 1998], gave a complete proof of 'Arnold's Theorem'
[V. I. Arnol'd. Small denominators and problems of stability of motion in
classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963),
91-192] on the planetary many-body problem, establishing, in particular, the
existence of a positive measure set of smooth (C\infty) Lagrangian invariant
tori for the planetary many-body problem. Here, using R\"u{\ss}mann's 2001 KAM
theory [H. R\"u{\ss}mann. Invariant tori in non-degenerate nearly integrable
Hamiltonian systems. R. & C. Dynamics 2(6) (2001), 119-203], we prove the above
result in the real-analytic class
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