Chaos in the dynamics generated by sequences of maps, with applications to chaotic advection in flows with aperiodic time dependence

Abstract

In this paper we develop analytical techniques for proving the existence of chaotic dynamics in systems where the dynamics is generated by infinite sequences of maps. These are generalizations of the Conley-Moser conditions that are used to show that a (single) map has an invariant Cantor set on which it is topologically conjugate to a subshift on the space of symbol sequences. The motivation for developing these methods is to apply them to the study of chaotic advection in fluid flows arising from velocity fields with aperiodic time dependence, and we show how dynamics generated by infinite sequences of maps arises naturally in that setting. Our methods do not require the existence of a homoclinic orbit in order to conclude the existence of chaotic dynamics. This is important for the class of fluid mechanical examples considered since one cannot readily identify a homoclinic orbit from the structure of the equations.¶We study three specific fluid mechanical examples: the Aref blinking vortex flow, Samelson's tidal advection model, and Min's rollup-merge map that models kinematics in the mixing layer. Each of these flows is modelled as a type of "blinking flow", which mathematically has the form of a linked twist map, or an infinite sequence of linked twist maps. We show that the nature of these blinking flows is such that it is possible to have a variety of "patches" of chaos in the flow corresponding to different length and time scales.