We study the ordinary differential equation εx ̈ + x ̇ + εg(x) = εf(ωt), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ Rd. We show that if there exists c0 ∈ R 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.

CORSI L, FEOLA R, & GENTILE G (2014). Convergent series for quasi-periodically forced strongly dissipative systems. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 16(3), 1350022 [10.1142/S0219199713500223].

Convergent series for quasi-periodically forced strongly dissipative systems

CORSI L;FEOLA R;GENTILE, Guido
2014

Abstract

We study the ordinary differential equation εx ̈ + x ̇ + εg(x) = εf(ωt), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ Rd. We show that if there exists c0 ∈ R 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/140344
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