Topological chaos, braiding and bifurcation of almost-cyclic sets (original) (raw)

Using heteroclinic orbits to quantify topological entropy in fluid flows

Chaos (Woodbury, N.Y.), 2016

Topological approaches to mixing are important tools to understand chaotic fluid flows, ranging from oceanic transport to the design of micro-mixers. Typically, topological entropy, the exponential growth rate of material lines, is used to quantify topological mixing. Computing topological entropy from the direct stretching rate is computationally expensive and sheds little light on the source of the mixing. Earlier approaches emphasized that topological entropy could be viewed as generated by the braiding of virtual, or "ghost," rods stirring the fluid in a periodic manner. Here, we demonstrate that topological entropy can also be viewed as generated by the braiding of ghost rods following heteroclinic orbits instead. We use the machinery of homotopic lobe dynamics, which extracts symbolic dynamics from finite-length pieces of stable and unstable manifolds attached to fixed points of the fluid flow. As an example, we focus on the topological entropy of a bounded, chaotic,...

Topological chaos and periodic braiding of almost-cyclic sets

2011

In certain (2+1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that ''stir''the surrounding domain and serve as the basis for this topological analysis.

Topological aspects of quantum chaos

Chaos: An Interdisciplinary Journal of Nonlinear Science, 1992

A full computation-relevant topological dynamics classification of elementary cellular automata Chaos 22, 043143 Rigorous results for the minimal speed of Kolmogorov-Petrovskii-Piscounov monotonic fronts with a cutoff Quantized classically chaotic maps on a toroidal two-dimensional phase space are studied. A discrete, topological criterion for phase-space localization is presented. To each eigenfunction is associated an integer, analogous to a quantized Hall conductivity, which tests the way the eigenfunction explores the phase space as some boundary conditions are changed. The correspondence between delocalization and chaotic classical dynamics is discussed, as well as the role of degeneracies of the eigenspectrum in the transition from localized to delocalized states. The general results are illustrated with a particular model.

Noise-driven topological changes in chaotic dynamics

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2021

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies-a technique originally introduced to characterize the topological structure of deterministically chaotic flows-which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points.

Ensemble-based topological entropy calculation (E-tec)

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019

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.

Generating topological chaos in lid-driven cavity flow

Physics of Fluids, 2007

Periodic motion of three stirrers in a two-dimensional flow can lead to chaotic transport of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of that motion and continuity of the fluid. Work in this area has focused largely on using physical rods as stirrers, but the theory also applies when the "stirrers" are passive fluid particles. We demonstrate the occurrence of topological chaos for Stokes flow in a two-dimensional lid-driven cavity without internal rods. This approach to stirring can enhance mixing relative to a "standard" chaos-generating lid-driven cavity flow.

Topological Effects of Noise on Nonlinear Dynamics

arXiv: Chaotic Dynamics, 2020

Noise modifies the behavior of chaotic systems. Algebraic topology sheds light on the most fundamental effects involved, as illustrated here by using the Lorenz (1963) model. This model's attractor is "strange" but frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Here, we use Branched Manifold Analysis via Homologies (BraMAH) to describe LORA's coarse-grained branched manifold. BraMAH is thus extended from deterministic flows to noise-driven systems. LORA's homology groups change in time and differ from the deterministic one.

Chaotic escape from an open vase-shaped cavity. II. Topological theory

Physical Review E, 2012

We present part II of a study of chaotic escape from an open two-dimensional vase-shaped cavity. A surface of section reveals that the chaotic dynamics is controlled by a homoclinic tangle, the union of stable and unstable manifolds attached to a hyperbolic fixed point. Furthermore, the surface of section rectifies escape-time graphs into sequences of escape segments; each sequence is called an epistrophe. Some of the escape segments (and therefore some of the epistrophes) are forced by the topology of the dynamics of the homoclinic tangle. These topologically forced structures can be predicted using the method called homotopic lobe dynamics (HLD). HLD takes a finite length of the unstable manifold and a judiciously altered topology and returns a set of symbolic dynamical equations that encode the folding and stretching of the unstable manifold. We present three applications of this method to three different lengths of the unstable manifold. Using each set of dynamical equations, we compute minimal sets of escape segments associated with the unstable manifold, and minimal sets associated with a burst of trajectories emanating from a point on the vase's boundary. The topological theory predicts most of the early escape segments that are found in numerical computations.