Chaos in differential equations driven by a nonautonomous force

Nonautonomous forces appear in many applications. They could be periodic, quasiperiodic and almost periodic in time; or they could take the form of a sample path of a random forcing driven by a stochastic process, which is without any periodicity in time. In this paper, we study the chaotic behaviour of differential equations driven by a general nonautonomous forcing without assuming any periodicity in time, aiming at applications to systems driven by a bounded random force. As a direct application, we prove that, for the Duffing equation driven by a bounded stationary stochastic process induced by a Brownian motion, chaotic dynamics exist almost surely. We also obtain various chaotic behaviour that are exclusively associated with equations driven by nonautonomous forcing without any periodicity in time. It has turned out that, unlike the systems driven by a periodic or almost periodic forcing, the transversal intersections of the stable and unstable manifolds are neither necessary nor sufficient for chaotic dynamics to exist. Finally, we apply all our results to the Duffing equation.