FM 011-2012 (arXiv:1211.2125; mp_arc 12-140)

Author: Livia Corsi, Roberto Feola and Guido Gentile

Title: Convergent series for quasi-periodically forced strongly dissipative systems
Abstract: We study the ordinary differential equation ε x'' + x' + ε g(x) = ε f(ω t), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ Rd. We show that if there exists c0R such that g(c0) equals the average of f and the first non-zero derivative of g at c0 is of odd order n, then, for ε small enough and under very mild Diophantine conditions on ω, there exists a quasi-periodic solution close to c0 with the same frequency vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on ω can be completely removed. This extends results previously available in the literature for n=1. We also point out that, if n=1 and the first derivative of g at c0 is positive, then the quasi-periodic solution is locally unique and attractive.

Keywords: Strongly dissipative systems, quasi-periodic forcing, quasi-periodic solutions, asymptotic expansion, non-degeneracy, irrationality conditions.

Livia Corsi
Dipartimento di Matematica
Università di Napoli "Federico II"
Monte Sant'Angelo, Via Cinthia, 80126 Napoli, Italy

Roberto Feola
Dipartimento di Matematica
Università di Roma "La Sapienza"
Piazzale Aldo Moro 5, 00185 Roma, Italy

Guido Gentile
Dipartimento di Matematica
Università di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy