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
e-mail: livia.corsi@unina.it

Roberto Feola
Dipartimento di Matematica
Università di Roma "La Sapienza"
Piazzale Aldo Moro 5, 00185 Roma, Italy
e-mail: feola@mat.uniroma1.it

Guido Gentile
Dipartimento di Matematica
Università di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it