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Quantifying chaos using Lagrangian descriptors.

M Hillebrand1, S Zimper1, A Ngapasare1

  • 1Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa.

Chaos (Woodbury, N.Y.)
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Summary
This summary is machine-generated.

We developed new methods using Lagrangian descriptors (LDs) to efficiently estimate chaos in dynamical systems. These techniques accurately distinguish chaotic from regular orbits, offering a faster alternative to existing methods.

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Area of Science:

  • Dynamical Systems Theory
  • Chaos Theory
  • Computational Physics

Background:

  • Estimating the chaoticity of orbits is crucial for understanding conservative dynamical systems.
  • Traditional methods for chaos detection can be computationally intensive.

Purpose of the Study:

  • To present and validate simple, efficient methods for quantifying chaos in low-dimensional conservative dynamical systems.
  • To introduce new indicators based on Lagrangian Descriptors (LDs) for chaos estimation.

Main Methods:

  • Computation of Lagrangian Descriptors (LDs) for orbits in Hamiltonian systems and symplectic maps.
  • Development of two new quantities: the difference and ratio of neighboring orbits' LDs.
  • Validation against the Smaller Alignment Index (SALI) method using Hénon-Heiles and standard map systems.

Main Results:

  • The proposed LD-based indicators accurately characterize orbit nature with >90% agreement with SALI.
  • Short-time, coarse-grid LD computations are sufficient for reliable chaos quantification.
  • The methods require less CPU time compared to SALI.

Conclusions:

  • Lagrangian Descriptors provide a computationally efficient and accurate tool for quantifying chaos in dynamical systems.
  • The developed indicators reveal local and global chaotic phase space structures.
  • LDs are suitable for investigating chaos in both continuous and discrete low-dimensional conservative systems.