Ensemble-based topological entropy calculation (E-tec)

Abstract

Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how trajectories "braid" around one another. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation, or E-tec, a method to derive a lower-bound on topological entropy of two-dimensional systems by considering the evolution of a "rubber band" (piece-wise linear curve) wrapped around the data points and evolving with their trajectories. The topological entropy is bounded below by the exponential growth rate of this band. We use tools from computational geometry to track the evolution of the rubber band as data points strike and deform it. Because we maintain information about the configuration of trajectories with respect to one another, updating the band configuration is performed locally, which allows E-tec to be more computationally efficient than some competing methods. In this work, we validate and illustrate many features of E-tec on a chaotic lid-driven cavity flow. In particular, we demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time.arXiv:1809.01005v3 [physics.comp-ph]

Topological chaos in active nematics

“…However, the original defect trajectories must be extended to exist for all times, as described in Supplementary Section S3. We use the recently developed E-tec algorithm [38] to extract topological entropy from defect trajectories. This algorithm uses a computational geometry approach to propagate an initial piecewise linear elastic mesh ( Fig.…”

“…Again, such ensembles typically provide only a lower bound on the true entropy of the flow-the more trajectories included, the greater the lower bound. In some special cases, a small number of specifically chosen trajectories generate all of the topological entropy [38]. Figure 3d shows three results, corresponding to stirring by just the +1/2 defects, by just the -1/2 defects, and by both +1/2 and -1/2 defects.…”

“…In fact, any collection of passively advected orbits in the fluid, not just those next to the stirring rods, generates a braid type, with an associated minimal topological entropy; such trajectories have been described as "ghost rods" in the literature [33,34]. Though the mathematics is more rigorous when such trajectories are periodic in time, recent work has generalized the analysis to open, aperiodic trajectories [35][36][37][38].…”

Using braids to quantify interface growth and coherence in a rotor-oscillator flow

“…Following Boyland et al [55], the study of the topological properties of fluid stirring has developed into an active research area. The topological approach is particularly well suited to the study of mixing by rods, vortices, or otherwise distinguished Lagrangian trajectories [56][57][58][59][60][61][62][63][64]. In particular Thiffeault [56,65], Allshouse and Thiffeault [49], Budišić and Thiffeault [66] used braid theory to characterize mixing and coherent structures from planar flows solely from particle trajectories, forming the basis for the approaches used in this paper.…”

“…The FTBE provides a robust measure of mixing that approaches the topological entropy as the number of trajectories is increased. More recently, Roberts et al [64] developed a braid-free approach that also estimates topological entropy based on the relationship with growth of material lines, without detailed knowledge of the velocity field.…”