## Convergent series and domains of analyticity for response solutions in quasi-periodically forced strongly dissipative systems

Series:
CDSNS Colloquium
Monday, March 25, 2013 - 16:05
1 hour (actually 50 minutes)
Location:
Skiles 006
,
University of Naples Federico II''
,
We study the ordinary differential equation \varepsilon \ddot x + \dot x + \varepsilon g(x) = \e f(\omega t), with f and g analytic and f quasi-periodic in t with frequency vector \omega\in\mathds{R}^{d}. We show that if there exists c_{0}\in\mathds{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 \mathfrak{n}, then, for \varepsilon small enough and under very mild Diophantine conditions on \omega, there exists a quasi-periodic solution "response 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. Moreover we show that for \mathfrak{n}=1 such a solution depends analytically on \e in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin. These results have been obtained in collaboration with Roberto Feola (Universit\a di Roma La Sapienza'') and Guido Gentile (Universit\a di Roma Tre).