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Universal finite-sample effect on the perturbation growth in chaotic dynamical systems
Hiroya Nakao1, Shuya Kitada, Alexander S Mikhailov
1Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. nakao@ton.scphys.kyoto-u.ac.jp
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 10, 2006
Summary
Finite sample sizes in chaotic dynamical systems simulations cause moment growth to relax from exponential. This study estimates relaxation time and reveals universal post-relaxation growth laws reflecting chaotic expansion rates.
Area of Science:
- * Chaos theory
- * Statistical mechanics
- * Computational physics
Background:
- * Numerical simulations of chaotic dynamical systems often encounter finite-sample effects.
- * The growth of perturbation moments in these systems is crucial for understanding system dynamics.
- * Limited sample trajectories in simulations lead to deviations from pure exponential growth.
Purpose of the Study:
- * To investigate the finite-sample effect on the growth of perturbation moments in chaotic systems.
- * To estimate the relaxation time and derive the post-relaxation growth law for these moments.
- * To demonstrate the universal nature of post-relaxation growth and its connection to chaotic expansion rates.
Main Methods:
- * Application of the large-deviation formalism to chaotic time series.
- * Numerical estimation of moments using a finite number of sample trajectories.
- * Derivation of analytical expressions for relaxation time and post-relaxation growth.
Main Results:
- * Finite sample sets cause a transition from initial pure exponential growth to relaxed growth of moments.
- * Rare events become unobservable in finite samples, leading to this transition.
- * Each moment follows a universal growth law even after relaxation, dependent on chaotic expansion statistics.
Conclusions:
- * Finite-sample effects significantly alter moment growth in chaotic system simulations.
- * The derived post-relaxation growth laws provide insights into the statistical properties of chaotic expansion.
- * Understanding these effects is crucial for accurate numerical analysis of chaotic dynamics.
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