On the dynamics of a time-periodic equation

Abstract

In this paper we use the second order equation $\frac{d^2 q}{dt^2} + (\lambda - \gamma q^2) \frac{d q}{dt} - q + q^3 = \mu q^2 \sin \omega t$ as a demonstrative example to illustrate how to apply the analysis of \cite{WO} and \cite{WOk} to the studies of concrete equations. We prove, among many other things, that there are positive measure sets of parameters $(\lambda, \gamma, \mu, \omega)$ corresponding to the case of intersected and the case of separated stable and unstable manifold of the solution $q(t) = 0$, $t \in \mathbb R$ respectively, so that the corresponding equations admit strange attractors with SRB measures.