12-120 Livia Corsi, Roberto Feola, Guido Gentile

Domains of analyticity for response solutions in strongly dissipative forced systems (211K, pdf) Oct 15, 12

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Abstract. We study the analyticity properties on $arepsilon$ of the ordinary differential equation $arepsilon\ddot x + \dot x + arepsilon g(x) = \e f(\omega t)$, where $g$ and $f$ are real-analytic functions, with $f$ quasi-periodic in $t$ with frequency vector $\omega$. A response solution is a quasi-periodic solution to the equation with the same frequency vector as the forcing. If $c_{0} \in \mathbb{R}$ is such that $g(c_0)$ equals the average of $f$ and $g'(c_0) eq0$, under very mild assumptions on $\omega$ there exists a response solution close to $c_0$. In particular no assumption at all is required on $\omega$ if $f$ is a trigonometric polynomial. We show that such a response solution depends analytically on $arepsilon$ in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin.

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