Convergent series for quasi-periodically forced strongly dissipative systems

TL;DR

This paper proves the existence of quasi-periodic solutions in strongly dissipative systems with quasi-periodic forcing, extending previous results to higher-order cases and removing Diophantine conditions when the forcing is a polynomial.

Contribution

It generalizes existing results by establishing quasi-periodic solutions for higher-order derivatives and removes Diophantine conditions for polynomial forcing.

Findings

Existence of quasi-periodic solutions under mild conditions.

Extension of results to higher-order derivatives n>1.

Removal of Diophantine conditions for polynomial forcing.

Abstract

We study the ordinary differential equation ${\varepsilon}\ddot x+\dot x + {\varepsilon} g(x) = {\varepsilon} f(\omega t)$, with $f$ and $g$ analytic and $f$ quasi-periodic in $t$ with frequency vector $\omega\in R^{d}$. We show that if there exists $c_0\in R$ such that $g(c_0)$ equals the average of $f$ and the first non-zero derivative of $g$ at $c_0$ is of odd order $n$, then, for ${\varepsilon}$ small enough and under very mild Diophantine conditions on $\omega$, there exists a quasi-periodic solution close to $c_0$, with the same frequency vector as $f$. In particular if $f$ is a trigonometric polynomial the Diophantine condition on $\omega$ can be completely removed. This extends results previously available in the literature for $n=1$. We also point out that, if $n=1$ and the first derivative of $g$ at $c_0$ is positive, then the quasi-periodic solution is locally unique and attractive.