On the dynamics of a time-periodic equation

TL;DR

This paper analyzes a specific time-periodic second-order differential equation, demonstrating the existence of strange attractors with SRB measures for certain parameter sets using advanced analytical methods.

Contribution

It applies and illustrates the analysis techniques of previous works to a concrete nonlinear equation, establishing conditions for strange attractors and manifold intersections.

Findings

Existence of positive measure parameter sets with intersected stable and unstable manifolds.

Presence of strange attractors with SRB measures under certain parameters.

Application of analytical methods to a specific nonlinear 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.