Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders

Abstract

In the present paper we prove a form of Arnold diffusion. The main result says that for a "generic" perturbation of a nearly integrable system of arbitrary degrees of freedom $n\ge 2$ \[ H_0(p)+\eps H_1(þ,p,t),\quad þ\in \T^n,\ p\in B^n,\ t\in \T=\R/\T, \] with strictly convex $H_0$ there exists an orbit $(þ_{\e},p_{e})(t)$ exhibiting Arnold diffusion in the sens that [\sup_{t>0}\|p(t)-p(0) \| >l(H_1)>0] where $l(H_1)$ is a positive constant independant of $\e$. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders of limited regularity, but of large size, extrapolating on \cite{Be3} and \cite{KZZ}. Once these cylinders are constructed we use versions of Mather variational method developed in Bernard \cite{Be1}, Cheng-Yan \cite{CY1, CY2}.

References in corpus (1)

• Normally hyperbolic invariant manifolds near strong double resonance

Cited by in corpus (13)

• Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories
• Arnold diffusion for a complete family of perturbations
• The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks
• Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom
• A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems
• Normally hyperbolic invariant manifolds near strong double resonance
• Normally Hyperbolic Invariant Laminations and diffusive behaviour for the generalized Arnold example away from resonances
• Arnold diffusion of charged particles in ABC magnetic fields
• Orbits of nearly integrable systems accumulating to KAM tori
• Diffusion along chains of normally hyperbolic cylinders
• Random Iteration of Cylinder Maps and diffusive behavior away from resonances
• A new measure of instability and topological entropy of area-preserving twist diffeomorphisms
• On the Existence of Drifting Orbits for Non-Convex Hamiltonian Systems