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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.
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.
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.
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