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

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 Poincaré index (a topological invariant of smooth systems) is not preserved.

References in corpus (4)

• Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua
• A Study in Three-Dimensional Chaotic Dynamics: Granular Flow and Transport in a Bi-Axial Spherical Tumbler
• Hamiltonian mappings and circle packing phase spaces: numerical investigations
• Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations

Cited by in corpus (1)

• A Unified Theory for Chaotic Mixing in Porous Media: from Pore Networks to Granular Systems