Convergent series for quasi-periodically forced strongly dissipative systems
We study the ordinary differential equation ${\varepsilon}\ddot x+\dot x + {\varepsilon} g(x) = {\varepsilon} f(ωt)$, with $f$ and $g$ analytic and $f$ quasi-periodic in $t$ with frequency vector $ω\in R^{d}$. 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 $ω$, 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 $ω$ 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.