Related papers: A Modified Parameterization Method for Invariant L…

In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian vector field having an invariant torus supporting…

In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are…

In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov…

Consider a real--analytic nearly--integrable mechanical system with potential $f$, namely, a Hamiltonian system, having a real-analytic Hamiltonian $$ H(y,x)=\frac12 | y |^2 +\e f(x)\ , $$ $y,x$ being $n$--dimensional standard action--angle…

In present paper, from the viewpoint of physical intuition we introduce a Hamiltonian system with multiscale rotation, which describes many systems, for example, the forced pendulum with fast rotation, weakly coupled $N$-oscillators with…

We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a…

The goal of this paper is to provide a methodology to prove existence of (fiberwise hyperbolic) real-analytic invariant tori in real-analytic quasi-periodic skew-product dynamical systems that present nearly-invariant tori of the same…

We consider autonomous Hamiltonian systems and present an algorithm to compute at the same time partially hyperbolic invariant tori (whiskered tori), as well as high-order expansions of their stable and unstable manifolds. Such whiskered…

We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in the sense that nearby solutions remain close to the torus…

In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In…

It is widespread since the beginning of KAM Theory that, under "sufficiently small" perturbation, of size $\epsilon$, apart a set of measure $O(\sqrt{\epsilon})$, all the KAM Tori of a non-degenerate integrable Hamiltonian system persist up…

We show that an analytic invariant torus $\cT_0$ with Diophantine frequency $\o_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\cT_0$ satisfies a R\"ussmann transversality…

The goal of this paper is to present a methodology for the computation of invariant tori in Hamiltonian systems combining flow map methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of…

This paper investigates in depth how stochastic perturbations affect the integrable structure of Hamiltonian systems and develops a KAM theory for stochastic Hamiltonian dynamics, in the sense of the most probable path. We first derive the…

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in \ { \mathbb{T}}^d, \qquad…

In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + g(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in {\mathbb T}^d, \qquad \qquad (*) $$ where $g(x,u)=4u^3+ O(u^4)$. Namely,…

We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the…

In this paper, we present an algorithm to computea fiberwise Lagrangian torus in quasi-periodic (QP) Hamiltonian systems, whose convergence is proved in the [CHP25]. We exhibit the algorithm with two models. The first is a Tokamak model…

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 $O(\sqrt{\varepsilon})$, if $\varepsilon$ is the size of…

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying R\"{u}ssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of…