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Finding Birkhoff averages via adaptive filtering.

M Ruth1, D Bindel2

  • 1Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.

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Summary
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We introduce Birkhoff reduced rank extrapolation (RRE) to classify trajectories of symplectic maps. This method efficiently identifies invariant tori and islands, distinguishing them from chaotic behavior.

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Area of Science:

  • Dynamical systems theory
  • Computational physics
  • Chaos theory

Background:

  • Classifying trajectories in symplectic maps is crucial for understanding dynamical systems.
  • Distinguishing between invariant tori, islands, and chaotic regions is a key challenge.

Purpose of the Study:

  • To develop a novel method for efficient trajectory classification in symplectic maps.
  • To utilize convergence properties for distinguishing invariant structures from chaos.

Main Methods:

  • Introduction of Birkhoff reduced rank extrapolation (RRE), a modified RRE technique.
  • Leveraging the convergence rate of Birkhoff RRE for trajectory classification.
  • Applying the method to analyze the standard map and magnetic field line dynamics.

Main Results:

  • Birkhoff RRE efficiently obtains ergodic averages for invariant tori and islands.
  • Birkhoff RRE exhibits slow convergence in chaotic regions, enabling classification.
  • The method successfully determines the number of islands and rotation numbers for invariant circles.

Conclusions:

  • Birkhoff RRE provides an efficient tool for classifying symplectic map trajectories.
  • It enables the parameterization of invariant circles and islands from single trajectories.
  • This method has practical applications in analyzing complex dynamical systems.