Domains of analyticity for response solutions in strongly dissipative forced systems

TL;DR

This paper investigates the analyticity of response solutions in strongly dissipative forced systems, demonstrating that under mild conditions, solutions depend analytically on the dissipation parameter in a domain tangent to the imaginary axis.

Contribution

It establishes the existence and analyticity of quasi-periodic solutions in a class of strongly dissipative differential equations with analytic forcing.

Findings

Response solutions exist near specific equilibrium points.

Solutions depend analytically on the dissipation parameter in a complex domain.

The domain of analyticity is tangent more than quadratically to the imaginary axis.

Abstract

We study the ordinary differential equation $\varepsilon\ddot x + \dot x + \varepsilon g(x) = \e f(\omega t)$, where $g$ and $f$ are real-analytic functions, with $f$ quasi-periodic in $t$ with frequency vector $\omega$. If $c_{0} \in \mathbb{R}$ is such that $g(c_0)$ equals the average of $f$ and $g'(c_0)\neq0$, under very mild assumptions on $\omega$ there exists a quasi-periodic solution close to $c_0$. We show that such a solution depends analytically on $\varepsilon$ in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin.