Globally and locally attractive solutions for quasi-periodically forced systems

We consider a class of differential equations, ¨x + γ ˙x + g(x) = f (ωt), with ω ∈ Rd , describing onedimensional 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) = x2p+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) = x2 (describing the varactor equation), we find that there is at least one trajectory which describes a local attractor. © 2006 Elsevier Inc. All rights reserved