Renato Calleja, Alessandra Celletti, and Rafael de la Llave
Construction of response functions in forced strongly dissipative systems
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ABSTRACT. We study the existence of quasi--periodic solutions $x$ of
the equation
$$arepsilon \ddot x + \dot x + arepsilon g(x) = arepsilon f(\omega t)\ ,$$
where $x: \mathbb{R}
ightarrow \mathbb{R}$ is the unknown and we are
given $g:\mathbb{R}
ightarrow \mathbb{R}$, $f: \mathbb{T}^d
ightarrow \mathbb{R}$, $\omega \in \mathbb{R}^d$. We assume that there is a $c_0\in \mathbb{R}$ such that $g(c_0) = \hat f_0$ (where $\hat f_0$ denotes the average of $f$) and $g'(c_0)
e 0$. Special cases of this equation, for example when $g(x)=x^2$, are called the "varactor problem" in the literature.
We show that if $f$, $g$ are analytic, and $\omega$ satisfies some
very mild irrationality conditions, there are families of quasi-periodic solutions with frequency $\omega$. These families depend analytically on $arepsilon$, when $arepsilon$ ranges over a complex domain that includes cones or parabolic domains based at the origin.
The irrationality conditions required in this paper are very weak.
They allow that the small denominators $|\omega