Topological Entropy and Secondary Folding

Abstract

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.

Estimating Topological Entropy from the Motion of Stirring Rods

“…The conjecture is that this flip causes secondary folding by promoting 'slack' in the material lines. This is evident when examining toral linked twist maps [28,29]. Unfortunately, this cannot be the whole story, since repeating the rod motion twice will always make the homological eigenvalue positive, but will clearly not make the lower bound any better.…”

“…A negative eigenvalue corresponds to a 'flip' of the homological generators at every application of the braid. For toral linked twist maps, this is associated with 'kinks' in the material lines, as shown in [28]. These are what we call 'secondary folds,' as depicted for a fluid system in Figure 5 and discussed in Section 3.1).…”

“…In other words, there must be some extra folding that is not directly due to the rods. We call this secondary folding [28,29]. Figure 5(a) shows an example of folding due to a rod, and Figure 5…”

“…At the heart of the matter is 'secondary folding,' or the observation that in some cases material lines fold a lot more than is strictly required by the topology (a) of the rod motion. This issue was explored in detail by the authors for toral linked twist maps [28,29]. Here we focus on physical rod-stirring devices, also called rod mixers.…”

CFD Based Qualification of Mixing Efficiency of Stirred Vessels

“…The breaking-up of the fluid reduces the size of the clumps, while the interdiffusion tends to equalize the concentration differences between the neighbouring regions of the mixture. Tumasz and Thiffeault created a "topological braiding exponent" to characterize the complexity of the braid generated by the motion of three or more fluid particles [8]. Another method for quantifying mixing is MixNorm, which has been created by Mathew et al [9].…”