Domains of analyticity and Gevrey estimates of tori in weakly dissipative systems

Abstract

We consider the problem of following quasi-periodic tori in perturbations of Hamiltonian systems which involve friction and external forcing.

In the first part, we study a family of dissipative standard maps of the cylinder for which the dissipation is a function of a small complex parameter of perturbation. We compute perturbative expansions formally in the parameter of perturbation and use them to estimate the shape of the domains of analyticity of invariant circles as functions of the parameter of perturbation. We also give evidence that the functions might belong to a Gevrey class. The numerical computations we perform support conjectures on the shape of the domains of analyticity.

In the second part, we consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. We prove that the formal expansions for the quasiperiodic solutions and the adjustable parameter satisfy Gevrey estimates. To prove this result we introduce a novel method that might be of interest beyond the problem considered in this work.