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Finite-time braiding exponents.

Marko Budišić1, Jean-Luc Thiffeault1

  • 1Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA.

Chaos (Woodbury, N.Y.)
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Summary
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We introduce a new method to calculate topological entropy using sparse flow trajectories represented as braids. This Finite-Time Braiding Exponent (FTBE) offers a robust alternative for complex fluid dynamics analysis.

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Area of Science:

  • Fluid dynamics
  • Dynamical systems theory
  • Chaos theory

Background:

  • Topological entropy quantifies chaos in dynamical systems.
  • Traditional methods require detailed velocity fields, often unavailable in sparse measurements.
  • Ocean flows and other complex systems present measurement challenges.

Purpose of the Study:

  • To develop a novel method for calculating topological entropy from sparse trajectory data.
  • To introduce the Finite-Time Braiding Exponent (FTBE) as a computable proxy for topological entropy.
  • To assess the robustness and data efficiency of the FTBE method.

Main Methods:

  • Representing sparse flow trajectories using algebraic braids.
  • Advecting material curves and simplifying them to loop coordinates.
  • Calculating the exponential rate of loop stretching over finite time intervals (FTBE).
  • Numerical simulations of the Aref Blinking Vortex flow.

Main Results:

  • FTBEs approximate topological entropy from below with increasing trajectory data.
  • The method is robust to numerical parameters and initial conditions.
  • Trajectories can be reused, significantly reducing data requirements.
  • FTBEs show promise for ergodic, mixing systems.

Conclusions:

  • FTBE offers a practical alternative for estimating topological entropy in sparsely measured flows.
  • The method is computationally efficient and robust.
  • This approach enhances the analysis of complex dynamical systems with limited data.