Related papers: Optimal stability and instability results for a cl…
We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with…
In this article, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper upper bound on the speed of Arnold diffusion…
In this article, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper upper bound on the speed of Arnold diffusion…
We improve the global Nekhoroshev stability for analytic quasi-convex nearly integrable Hamiltonian systems. The new stability result is optimal, as it matches the fastest speed of Arnold diffusion.
Consider a sufficiently smooth nearly integrable Hamiltonian system of two and a half degrees of freedom in action-angle coordinates \[ H_\epsilon (\varphi,I,t)=H_0(I)+\epsilon H_1(\varphi,I,t), \varphi\in T^2,\ I\in U\subset R^2,\ t\in…
We prove the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic…
In the present paper we apply the geometrical mechanism of diffusion in an \emph{a priori} unstable Hamiltonian system with 3 $+$ 1/2 degrees of freedom. This mechanism consists of combining iterations of the \emph{inner} and \emph{outer}…
In this work we illustrate the Arnold diffusion in a concrete example---the \emph{a priori} unstable Hamiltonian system of $2+1/2$ degrees of freedom $H(p,q,I,\varphi,s) = p^{2}/2+\cos q -1 +I^{2}/2 + h(q,\varphi,s;\varepsilon)$---proving…
We expose some selected topics concerning the instability of the action variables in a priori unstable Hamiltonian systems, and outline a new strategy that may allow to apply these methods to a priori stable systems.
We consider the problem of Arnold's diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also develop a new method for measuring the…
We study the Arnold diffusion in a priori unstable near-integrable systems in a neighbourhood of a resonance of low order. We consider a non-autonomous near-integrable Hamiltonian system with $n+1/2$ degrees of freedom, $n\ge 2$. Let the…
It is well known that instabilities of nearly integrable Hamiltonian systems occur around resonances. Dynamics near resonances of these systems is well approximated by the associated averaged system, called slow system. Each resonance is…
We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special…
In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover,…
It is well known that under generic $C^r$ smooth perturbations, the phenomenon of global instability, known as Arnold diffusion, exists in a priori unstable Hamiltonian systems. In this paper, by using variational methods, we will prove…
Arnold's diffusion in quasi integrable hamiltonian systems occurs in exponentially large time. We study an initially hyperbolic system which admits diffusion in polynomial time.
This is a short survey on Nekhoroshev theory, KAM theory, and Arnold's diffusion.
In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$ H(x,y)=h(y)+\epsilon P(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3. $$ Under typical…
In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action…
We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) + $\epsilon$H1($\theta$, p, t), $\theta$ $\in$ T n , p $\in$ B n , t $\in$ T = R/T be a nearly integrable system of arbitrary degrees of freedom n 2 with a strictly…