Quasi-periodic motions in strongly dissipative forced systems

Abstract

We consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing.

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

“…Moreover, we also show that the results obtained here also apply to quasi-periodic perturbations of systems with dissipation, which have been considered in the literature ( [5,13,14]). However, our method does not require that the system is analytic or close to the integrable case (see Section 5).…”

“…The motivation is that quasi-periodic perturbations of conformally symplectic systems have been widely considered in the literature (see [5,13], where a one-dimensional dissipative system subject to a quasi-periodic forcing term has been considered).…”

Quasiperiodic motions in dynamical systems: Review of a renormalization group approach

“…Actually one can prove that M [n] (0, ε) is real for real ε [41], a property which becomes essential to deal with the case in which the nondegeneracy condition a = 0 is not satisfied -see Section 12.2 below. Again the properties (9.1) and (9.4) are proved together, by induction on n: for n = 0 they trivially hold, and, by assuming that they are satisfied up to n − 1, one sees that the series defining M [n] (x, ε) converge, and hence (9.1) can be proved for n; see [42,40,41] for details.…”

“…All the results of the previous sections can be extended to rotation vectors satisfying the Bryuno condition: see [39] for maximal and lower-dimensional tori, [9] for the standard map, and [40] for dissipative systems. For d = 2 one can write ω = (ω 1 , ω 2 ) = (1, α)ω 1 , where α = ω 2 /ω 1 is the rotation number.…”

“…Such a condition can be removed, and the existence of a response solution can be proved under the only condition that there exists a zero c 0 of odd order to the equation g(x) − f 0 = 0; see [40], where it is also proved that there is no response solution reducing to c 0 as ε → 0 if c 0 is a zero of even order.…”