Topological Chaos and Periodic Braiding of Almost-Cyclic Sets

Stremler, Mark A.; Ross, Shane D.; Grover, Piyush; Kumar, Pankaj

doi:10.1103/physrevlett.106.114101

Cited by 32 publications

(25 citation statements)

References 34 publications

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“…The method starts with computing Lagrangian trajectories for a number of initial positions, i.e., F τ t (x i ), i = 1, 2, · · · , N . The procedure P then consists of computing the topological entropy associated with combinations of these trajectories [92,93,94,95], and then separating trajectories into groups which have nearly identical entropy. Numerical application of this method requires that N be not large (∼ O(50)) to be able to address all possible combinations of trajectories.…”

Section: Lcs Diagnostic Methodsmentioning

confidence: 99%

Generalized Lagrangian coherent structures

Balasuriya

Ouellette

Rypina

2018

Physica D: Nonlinear Phenomena

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The notion of a Lagrangian Coherent Structure (LCS) is by now well established as a way to capture transient coherent transport dynamics in unsteady and aperiodic fluid flows that are known over finite time. We show that the concept of an LCS can be generalized to capture coherence in other quantities of interest that are transported by, but not fully locked to, the fluid. Such quantities include those with dynamic, biological, chemical, or thermodynamic relevance, such as temperature, pollutant concentration, vorticity, kinetic energy, plankton density, and so on. We provide a conceptual framework for identifying the Generalized Lagrangian Coherent Structures (GLCSs) associated with such evolving quantities. We show how LCSs can be seen as a special case within this framework, and provide an overarching discussion of various methods for identifying LCSs. The utility of this more general viewpoint is highlighted through a variety of examples. We also show that although LCSs approximate GLCSs in certain limiting situations under restrictive assumptions on how the velocity field affects the additional quantities of interest, LCSs are not in general sufficient to describe their coherent transport.

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Section: Lcs Diagnostic Methodsmentioning

confidence: 99%

Generalized Lagrangian coherent structures

Balasuriya

Ouellette

Rypina

2018

Physica D: Nonlinear Phenomena

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“…However, Figure 3 shows that h TN;3 is a lower bound for h f for approximately a 7% perturbation in s f . 41 In the discussion below, we consider the validity of the topological entropy predicted by TNCT even if we cannot identify exactly periodic orbits that produce the expected braiding motion. We accomplish this by using a transfer operator approach to reveal phase space structures that braid non-trivially and persist under perturbations in the s f values.…”

Section: B Perturbation Of the Reference Casementioning

confidence: 99%

“…The topological entropy given by the TNCT is no longer a strict lower bound, since the timeperiodic braid is only an approximation of the true dynamical structure. However, it was demonstrated 41 that this estimate can give a good representation of the flow.…”

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confidence: 99%

“…Our earlier work 41 considers creeping flow in a liddriven cavity and starts with parameters for which there exist three periodic orbits, which we refer to as the reference case. For small perturbations from the reference case, consideration of ACS braiding gives a clear explanation for the topological entropy in this flow.…”

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confidence: 99%

“…41 In this view, the domain is divided into distinct subsets such that there is a very small probability that typical trajectories beginning in each subset will leave this subset in a short time. These almost-invariant sets (AISs) 42 can be determined from the eigenspectrum of the discretized Perron-Frobenius transfer operator via a setoriented approach, which we review in Sec.…”

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confidence: 99%

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Topological chaos, braiding and bifurcation of almost-cyclic sets

Grover

Ross

Stremler

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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