Domains of Analyticity and Gevrey estimates in weakly dissipative systems.

Date:

Friday, August 27, 2021.

Time:

12:00pm - 2:00pm

Location:

Bluejeans meeting: https://bluejeans.com/417759047/0103

Committe members:

Dr. Rafael de la Llave (Advisor) - School of Mathematics, Georgia Institute of Technology

Dr. Chongchun Zeng - School of Mathematics, Georgia Institute of Technology

Dr. Molei Tao - School of Mathematics, Georgia Institute of Technology

Dr. Alex Blumenthal - School of Mathematics, Georgia Institute of Technology

Dr. Alessandra Celletti - Department of Mathematics, University of Rome Tor Vergata

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, $\varepsilon$.  We compute perturbative expansions formally in $\varepsilon$ and use them to estimate the shape of the domains of analyticity of invariant circles as functions of $\varepsilon$. 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 study rigorously the(divergent) series of formal expansions of the torus obtained using Lindstedt method.   We show that, for some systems in the literature, the series is Gevrey. We hope that the method of proof can be of independent interest: We develop KAM estimates for the divergent series. In contrast with the regular KAM method, we loose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey.