Globally and locally attractive solutions for quasi-periodically forced systems

Abstract

We consider a class of differential equations, $\ddot x + γ\dot x + g(x) = f(ωt)$, with $ω\in {\bf R}^{d}$, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of the limit cycle described by the trajectory with the same quasi-periodicity as the forcing. For $g(x)=x^{2p+1}$, $p\in {\bf N}$, we show that, when the dissipation coefficient is large enough, there is only one limit cycle and that it is a global attractor. In the case of other forces, including $g(x)=x^{2p}$ (with $p=1$ describing the varactor equation), we find estimates for the basin of attraction of the limit cycle.