The genericity of Arnold diffusion in nearly integrable Hamiltonian systems

Arnold Instability for Nearly{integrable Analytic Hamiltonian Systems

A notion of instability, which is proposed as a deenition for the so{called \Arnold diiusion", for one{parameter families of nearly{integrable analytic Hamiltonian systems, is introduced. An example of unstable system is given.

Arnold diffusion in perturbations of analytic integrable Hamiltonian systems

On the stability problem for nearly{integrable Hamiltonian systems

The problem of stability of the action variables in nearly{integrable (real{ analytic) Hamiltonian systems is considered. Several results (fully described in CG2]) are discussed; in particular: (i) a generalization of Arnold's method (A]) allowing to prove instability (i.e. drift of action variables by an amount of order 1, often called \Arnold's diiusion") for general perturbations of \a{prior...

Whiskered and Low Dimensional Tori in Nearly Integrable Hamiltonian Systems

We show that a nearly integrable hamiltonian system has invariant tori of all dimensions smaller than the number of degrees of freedom provided that certain nondegeneracy conditions are met. The tori we construct are generated by the resonances of the system and are topologically different from the orbits that are present in the integrable system. We also show that the tori we construct have st...

Prevalence of exponential stability among nearly-integrable Hamiltonian systems

In the 70’s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Recently, Guzzo has given examples of expon...