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Lyapunov exponents for Hamiltonian systems under small Lévy-type perturbations
Ying Chao1, Pingyuan Wei2, Jinqiao Duan3
1School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, P. R. China.
Chaos (Woodbury, N.Y.)
|September 2, 2021
Summary
This study estimates the Lyapunov exponent for Hamiltonian systems with non-Gaussian Lévy noise. The findings offer insights into the stability of dynamical systems under complex noise perturbations.
Area of Science:
- Dynamical Systems and Chaos Theory
- Stochastic Differential Equations
- Mathematical Physics
Background:
- Hamiltonian systems are fundamental in classical mechanics.
- Non-Gaussian Lévy-type noise introduces complex dynamics not captured by traditional Brownian motion.
- Understanding system stability under such noise is crucial for accurate modeling.
Purpose of the Study:
- To investigate the top Lyapunov exponent for Hamiltonian systems subjected to small non-Gaussian Lévy-type noise.
- To develop a method for estimating this exponent in systems with bounded jumps.
- To analyze the impact of non-Gaussian noise on system stability.
Main Methods:
- Linearization of the Hamiltonian system in a moving frame.
- Perturbation analysis of a nilpotent linear system.
- Application of the Pinsky-Wihstutz transformation.
- Utilizing the Khas'minskii formula for Lyapunov exponent estimation.
Main Results:
- An estimation method for the top Lyapunov exponent is established for the considered class of systems.
- The method is valid under assumptions of smoothness, ergodicity, and integrability.
- The theoretical framework is demonstrated with two illustrative examples.
Conclusions:
- The study provides a robust method for analyzing the stability of Hamiltonian systems with non-Gaussian Lévy noise.
- The findings contribute to the theoretical understanding of stochastic dynamical systems.
- The presented examples validate the applicability of the developed technique.
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