Topological chaos, braiding and bifurcation of almost-cyclic sets

Abstract

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or "ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems. When a body of fluid moves, whether it be in the atmosphere, an ocean, or a kitchen sink, there are often regions of fluid that move together for an extended period of time. As these coherent sets of fluid trace out trajectories in space and time, they can be thought of as "stirring" the surrounding fluid. The entanglement of these trajectories as they "braid" around each other is connected to the level of chaos present in the fluid system. When the sets return periodically to their initial positions in a twodimensional, time-dependent system, the entanglement of their trajectories can be used to predict a lower bound on the rate of stretching in the surrounding domain. We examine the trajectories of such "almost-cyclic sets" in a lid-driven cavity flow and demonstrate that this combination of topological analysis with set-oriented methods can be an effective means of predicting chaos. The characterization of the entanglement and associated prediction of stretching are achieved through application of the Thurston-Nielsen Classification Theorem, which in general classifies the topological complexity of homeomorphisms of punctured surfaces. While a rigorous lower bound on topological entropy is not available in the absence of exactly periodic braiding structures, our approach finds the "topological skeleton" that can be used to get an approximate rate of stretching.

Shape Coherence and Finite-Time Curvature Evolution

“…[1][2][3][4][5][6][7]. There have been a wide range of notions of coherence, from spectral, [8], to set oriented, [9] and through transfer operators [3,5] as well as variational principles [10], and even topological methods, [11,12]. Traditionally there has been an emphasis on vorticity [13], but generally an understanding that, coherent motions have a role in maintenance (production and dissipation) of turbulence in a boundary layer, [14].…”

Ensemble-based topological entropy calculation (E-tec)

“…The chaotic lid-driven cavity model 32,33,36,37 is a twodimensional area-preserving flow defined over a 2D vertical cross-section of a rectangular cavity, extending vertically from −b ≤ y ≤ b and horizontally from 0 ≤ x ≤ a. 3) with τ f = 0.96, guaranteeing the existence of a period-three orbit, seen in Fig 6c.…”

“…The stream function is an exact solution of the biharmonic equation ∇ 2 ∇ 2 ψ(x, y) = 0 defined on the rectangular domain. The stream function is time-periodic with period τ f and is given explicitly by We follow Grover et al 32 and assign U 1 = 9.92786, U 2 = 8.34932, a = 6, and b = 1. Fig.…”

Optimal transport over nonlinear systems via infinitesimal generators on graphs

“…We solve the the optimal transport problem by discretizing X into m = 60 × 30 boxes for different time horizons t f . For each t f , we choose k such that ∆t = t f k = 1 40 . We use a piecewise-constant approximation of the time-dependent drift vector field, i.e., g 0 (x, t j + δt) ≈ g 0 (x, t j ) ∀ δt ≤ ∆t, j ≤ k, for computing the corresponding generator using Eq.…”