12-79 Renato Calleja, Alessandra Celletti, and Rafael de la Llave

Construction of response functions in forced strongly dissipative systems (548K, pdf) Jul 27, 12

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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

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