Globally and locally attractive solutions for quasi-periodically forced systems

Abstract

We consider a class of differential equations, x ¨ + γ x ˙ + g ( x ) = f ( ω t ) , with ω ∈ R d , describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g ( x ) = x 2 p + 1 , p ∈ N , we show that, when the dissipation coefficient is large enough, there is only one such trajectory and that it describes a global attractor. In the case of more general nonlinearities, including g ( x ) = x 2 (describing the varactor equation), we find that there is at least one trajectory which describes a local attractor.