Mixing and transport from combined stretching-and-folding and cutting-and-shuffling

TL;DR

This paper investigates the complex interplay of stretching, folding, and cutting-and-shuffling in mixing systems, revealing unique bifurcations and non-mixing island behaviors in a granular tumbler model.

Contribution

It identifies and characterizes two novel bifurcations in systems combining stretching-and-folding with cutting-and-shuffling, expanding understanding of mixing dynamics.

Findings

Bifurcation of elliptic island containment by manifolds and cutting line tangency.

Maximum non-mixing region size occurs at the bifurcation point.

Periodic points can be annihilated by cutting-and-shuffling, breaking topological invariants.

Abstract

While structures and bifurcations controlling tracer particle transport and mixing have been studied extensively for systems with only stretching-and-folding, and to a lesser extent for systems with only cutting-and-shuffling, few studies have considered systems with a combination of both. We demonstrate two bifurcations for non-mixing islands associated with elliptic periodic points that only occur in systems with combined cutting-and-shuffling and stretching-and-folding, using as an example a map approximating biaxial rotation of a less-than-half-full spherical granular tumbler. First, we characterize a bifurcation of elliptic island containment, from containment by manifolds associated with hyperbolic periodic points to containment by cutting line tangency. As a result, the maximum size of the non-mixing region occurs when the island is at the bifurcation point. We also demonstrate a bifurcation where periodic points are annihilated by the cutting-and-shuffling action. Chains of elliptic and hyperbolic periodic points that arise when invariant tori surrounding an elliptic point break up [according to Kolmogorov-Arnold-Moser (KAM) theory] can annihilate when they meet a cutting line. Consequently, the Poincar\'{e} index (a topological invariant of smooth systems) is not preserved.