Computation of Domains of Analyticity for the dissipative standard map in the limit of small dissipation

TL;DR

This paper investigates the domains of analyticity for invariant circles in a family of conformally symplectic standard maps with small dissipation, using perturbative expansions and numerical methods to understand their structure.

Contribution

It provides a formal perturbative approach and numerical evidence for the shape of the analyticity domains of invariant circles in dissipative standard maps near zero dissipation.

Findings

Domains of analyticity are estimated using perturbative expansions.

Numerical continuation supports conjectures on the shape of these domains.

Functions may belong to a Gevrey class at zero dissipation.

Abstract

Conformally symplectic systems include mechanical systems with a friction proportional to the velocity. Geometrically, these systems transform a symplectic form into a multiple of itself making the systems dissipative or expanding. In the present work we consider the limit of small dissipation. The example we study is a family of conformally symplectic standard maps of the cylinder for which the conformal factor, $b(\varepsilon)$, is a function of a small complex parameter, $\varepsilon$. We assume that for $\varepsilon=0$ the map preserves the symplectic form and the dependence on $\varepsilon$ is cubic, i.e., $b(\varepsilon) = 1 - \varepsilon^3$. 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 at $\varepsilon = 0$. We also perform numerical continuation of the solutions as they pass through the boundary of the domain to illustrate that the monodromy of the solutions is trivial. The numerical computations we perform support conjectures on the shape of the domains of analyticity.