Georgia Institute of Technology College of Sciences

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).