63 research outputs found
Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians
In this paper, we give a new construction of resonant normal forms with a
small remainder for near-integrable Hamiltonians at a quasi-periodic frequency.
The construction is based on the special case of a periodic frequency, a
Diophantine result concerning the approximation of a vector by independent
periodic vectors and a technique of composition of periodic averaging. It
enables us to deal with non-analytic Hamiltonians, and in this first part we
will focus on Gevrey Hamiltonians and derive normal forms with an exponentially
small remainder. This extends a result which was known for analytic
Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As
applications, we obtain an exponentially large upper bound on the stability
time for the evolution of the action variables and an exponentially small upper
bound on the splitting of invariant manifolds for hyperbolic tori, generalizing
corresponding results for analytic Hamiltonians
Nekhoroshev's estimates for quasi-periodic time-dependent perturbations
In this paper, we consider a Diophantine quasi-periodic time-dependent
analytic perturbation of a convex integrable Hamiltonian system, and we prove a
result of stability of the action variables for an exponentially long interval
of time. This extends known results for periodic time-dependent perturbations,
and partly solves a long standing conjecture of Chirikov and Lochak. We also
obtain improved stability estimates close to resonances or far away from
resonances, and a more general result without any Diophantine condition
Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians
A major result concerning perturbations of integrable Hamiltonian systems is
given by Nekhoroshev estimates, which ensures exponential stability of all
solutions provided the system is analytic and the integrable Hamiltonian not
too degenerate. In the particular but important case where the latter is
quasi-convex, these exponential estimates have been generalized by Marco and
Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by
Lochak in the analytic case. In this paper, using the same approach we will
investigate the situation where the Hamiltonian is assumed to be only finitely
differentiable, it is known that exponential stability does not hold but
nevertheless we will prove estimates of polynomial stability
Generic super-exponential stability of invariant tori in Hamiltonian systems
In this article, we consider solutions starting close to some linearly stable
invariant tori in an analytic Hamiltonian system and we prove results of
stability for a super-exponentially long interval of time, under generic
conditions. The proof combines classical Birkhoff normal forms and a new method
to obtain generic Nekhoroshev estimates developed by the author and L.
Niederman in another paper. We will mainly focus on the neighbourhood of
elliptic fixed points, the other cases being completely similar
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