On the measure of the KAM-tori in a neighbourhood of a separatrix
A. G. Medvedev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Abstract:
Consider a Liouville-integrable Hamiltonian system with
$n$ degrees of freedom. Assume that the foliation of the phase space by invariant Lagrangian
$n$-tori is degenerate on a
$(2n-1)$-dimensional singular manifold
$\mathbb{W}$ formed by the asymptotic manifolds of hyperbolic
$(n-1)$-tori. The system usually ceases to be integrable after a small perturbation of order
$\varepsilon$, but in accordance with the KAM-theory most invariant
$n$-tori persist. The dynamics on the complement
$C$ to this toric set is commonly associated with chaos.
The measure of the set of points obtained as the intersection of a neighbourhood of
$\mathbb{W}$ with
$C$ is considered. Under natural assumptions it has the order of
$\sqrt \varepsilon$.
This results generalizes and complements the estimates for the measure of
$C$ away from
$\mathbb{W}$ due to Svanidze, Neishtadt and Pöschel.
Bibliography: 14 titles.
Keywords:
KAM-theory, separatrices, systems with small parameter, chaos, measure of the invariant tori, perturbation theory.
MSC: 37J40,
70H08 Received: 12.05.2023 and 12.03.2024
DOI:
10.4213/sm9955
Fulltext:
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