HAPPENING VIRTUALLY: 2021 SIAM Conference on Dynamical Systems (times listed are Eastern Time)

Abstract. A fundamental model for laminar fluid mixing by chaotic advection is that of orthogonal shears on the torus. Composing such shears gives an area-preserving, discrete time map, modelling the iterative process of stretching and folding. The Arnold Cat Map is the canonical such example, and mixing properties can be proved straightforwardly. Here we present a new family of maps, incorporating a non-monotonic shear to more closely resemble realistic fluid velocity profiles, and prove mixing results for a wide parameter range.

In 2005 Cerbelli and Giona proved the mixing property for a particular case, focusing the first shear precisely half-way along the domain. This simplified the dynamics significantly, allowing for a direct approach. We build on their geometric arguments and follow a scheme from Katok and Strelcyn to extend the result to this broader family of non-monotonic maps. Importantly, this tells us that non-monotonic shears are not a barrier to analytical methods, only surmountable when the dynamics are simplified by some unique feature of the map. Aside from applications, our parameterised family is a rich source of varied dynamics; including the uniformly hyperbolic Arnold Cat Map, the non-uniformly hyperbolic (yet well behaved) map from Cerbelli and Giona, an entirely periodic map, and everything in between.