Dynamical cocycles with values in the Artin braid group

Gambaudo, Jean-Marc; Pécou, Elisabeth

doi:10.1017/s0143385799130207

Cited by 15 publications

(15 citation statements)

References 5 publications

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“…The American Physical Society trary particle orbits with braids group elements was first proposed by Gambaudo and Pécou [8].…”

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confidence: 99%

Measuring Topological Chaos

Thiffeault

2005

Phys. Rev. Lett.

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The orbits of fluid particles in two dimensions effectively act as topological obstacles to material lines. A spacetime plot of the orbits of such particles can be regarded as a braid whose properties reflect the underlying dynamics. For a chaotic flow, the braid generated by the motion of three or more fluid particles is computed. A "braiding exponent" is then defined to characterize the complexity of the braid. This exponent is proportional to the usual Lyapunov exponent of the flow, associated with separation of nearby trajectories. Measuring chaos in this manner has several advantages, especially from the experimental viewpoint, since neither nearby trajectories nor derivatives of the velocity field are needed.

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“…The American Physical Society trary particle orbits with braids group elements was first proposed by Gambaudo and Pécou [8].…”

mentioning

confidence: 99%

Measuring Topological Chaos

Thiffeault

2005

Phys. Rev. Lett.

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“…2(c). A simple method to do this was originally described in [7], but is also implicit in earlier work such as [18,25] (see also [26] for a related technique).…”

Section: B Extracting the Braid From A Flowmentioning

confidence: 99%

Braids of entangled particle trajectories

Thiffeault

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

128

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In many applications, the two-dimensional trajectories of fluid particles are available, but little is known about the underlying flow. Oceanic floats are a clear example. To extract quantitative information from such data, one can measure singleparticle dispersion coefficients, but this only uses one trajectory at a time, so much of the information on relative motion is lost. In some circumstances the trajectories happen to remain close long enough to measure finite-time Lyapunov exponents, but this is rare. We propose to use tools from braid theory and the topology of surface mappings to approximate the topological entropy of the underlying flow. The procedure uses all the trajectory data and is inherently global. The topological entropy is a measure of the entanglement of the trajectories, and converges to zero if they are not entangled in a complex manner (for instance, if the trajectories are all in a large vortex). We illustrate the techniques on some simple dynamical systems and on float data from the Labrador sea. The method could eventually be used to identify Lagrangian coherent structures present in the flow. Keywords: topological chaos, dynamical systems, Lagrangian coherent structuresConsider particles floating on top of a fluid. We can follow their trajectories, either with a camera or by computer simulation. If we then plot their position in a three-dimensional graph, with time the vertical coordinate, we get a 'spaghetti plot,' which contains information about how entangled the trajectories are. We discuss how to measure the level of entanglements in terms of topological entropy, and the interpretation of the results. This provides a straightforward method of estimating the level of chaos present in a system. This approach could also be used to determine if some trajectories remain together for a long time, and are thus part of a Lagrangian coherent structure.

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“…This work was motivated by the study of two-dimensional fluid mixing via the braiding motion of fluid particle trajectories [11,18]. In this setting we have derived an efficient tool that allows practical analysis of large braids.…”

Section: Calculating Topological Entropies From Laminationsmentioning

confidence: 99%

“…Investigation of two-dimensional fluid mixing by topological techniques is rapidly gaining popularity [1]. Topological perspectives on mixing either involve studying braiding motion of the stirring apparatus itself [5], or the diagnosis of mixing by analyzing braiding of orbits of the flow [10][11][12]18]. The quantity that is usually of interest is the topological entropy of the braid [4], which serves as a lower bound for the topological entropy of the flow.…”

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Topological Entropy of Braids on the Torus

Finn¹,

Thiffeault²

2007

SIAM J. Appl. Dyn. Syst.

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Abstract. A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir's technique for braids on the sphere. Previous methods for computing topological entropies include the Bestvina-Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina-Handel algorithm quickly becomes computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and appears to give exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow. The efficiency of the method allows us to explore how much extra information about flow entropy is encoded in the braid as the number of trajectories becomes large.

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Dynamical cocycles with values in the Artin braid group

Cited by 15 publications

References 5 publications

Measuring Topological Chaos

Measuring Topological Chaos

Braids of entangled particle trajectories

Topological Entropy of Braids on the Torus

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