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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

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Computing periodic orbits with arbitrary precision.

Alberto Abad¹, Roberto Barrio, Angeles Dena

¹Department of Theoretical Physics and IUMA, University of Zaragoza, Zaragoza, Spain. abad@unizar.es

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|August 27, 2011

Summary

This summary is machine-generated.

This study introduces a novel algorithm for precisely calculating periodic orbits in dynamical systems. The method achieves high-precision results for ordinary differential equations (ODEs), useful in physical system analysis.

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Area of Science:

Dynamical Systems and Chaos Theory
Numerical Analysis and Computational Science

Background:

Accurate computation of periodic orbits is crucial for understanding complex behaviors in physical systems.
Existing methods often lack the precision required for advanced analyses, such as complex pole location.

Purpose of the Study:

To develop and present a highly precise algorithm for computing periodic orbits of dynamical systems.
To demonstrate the algorithm's capability in achieving arbitrary precision, up to thousands of digits.

Main Methods:

The algorithm employs an optimized shooting method integrated with a Taylor-series based numerical ordinary differential equation (ODE) solver.
This combined approach is designed for high-precision computation of solutions to ODEs.

Main Results:

The developed methodology achieves quadratic convergence, enabling computations with precision up to thousands of digits.
Numerical tests on the Lorenz model and Hénon-Heiles Hamiltonian successfully generated periodic orbits with up to 1000-digit accuracy.

Conclusions:

The presented algorithm offers a robust and highly accurate solution for calculating periodic orbits in dynamical systems.
This advancement is particularly valuable for physical systems requiring extremely precise pole location analysis.