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Fast numerical test of hyperbolic chaos.
1Department of Instrumentation Engineering, Saratov State Technical University, Politekhnicheskaya 77, Saratov 410054, Russia. p.kuptsov@rambler.ru
Summary
This study introduces a fast numerical method to test chaotic dynamics hyperbolicity. The technique efficiently identifies tangencies between expanding and contracting subspaces, crucial for understanding complex systems.
Area of Science:
- Dynamical Systems and Chaos Theory
- Numerical Analysis and Computational Science
Background:
- Hyperbolicity is a fundamental property in analyzing the stability and predictability of dynamical systems, particularly chaotic ones.
- Traditional methods for testing hyperbolicity can be computationally intensive and require significant memory resources.
Purpose of the Study:
- To propose an effective and efficient numerical method for testing the hyperbolicity of chaotic dynamics.
- To develop a technique that avoids explicit computation of covariant Lyapunov vectors while still assessing system properties.
Main Methods:
- The proposed method adapts algorithms for covariant Lyapunov vectors, focusing on a characteristic value distribution.
- It involves solving equations for infinitesimal perturbations, with the number of equations related to positive and zero Lyapunov exponents.
- The method avoids direct calculation of covariant Lyapunov vectors, reducing computational load.
Main Results:
- The method yields a characteristic value distribution bounded within the unit interval.
- A zero value in this distribution signifies a tangency between expanding and contracting subspaces, indicating hyperbolicity.
- The number of required perturbation equations is typically much smaller than the full phase space dimension.
Conclusions:
- The suggested numerical method offers a fast and memory-efficient approach for testing hyperbolicity in chaotic systems.
- This technique provides a valuable tool for analyzing the geometric and dynamical properties of complex systems.
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