Lyapunov Exponents for Hamiltonian Systems under Small L\'evy Perturbations

TL;DR

This paper analyzes the Lyapunov exponent for Hamiltonian systems influenced by small non-Gaussian Le9vy noise, providing estimates of system stability and growth rates through advanced mathematical transformations.

Contribution

It introduces a novel approach to estimate Lyapunov exponents for Hamiltonian systems under Le9vy noise using Pinsky-Wihstutz transformation and Khas'minskii formula.

Findings

Lyapunov exponents characterized for systems with Le9vy noise.

Estimates of growth or decay rates under non-Gaussian perturbations.

Illustrative examples demonstrating the theoretical results.

Abstract

This work is to investigate the (top) Lyapunov exponent for a class of Hamiltonian systems under small non-Gaussian L\'evy noise. In a suitable moving frame, the linearisation of such a system can be regarded as a small perturbation of a nilpotent linear system. The Lyapunov exponent is then estimated by taking a Pinsky-Wihstutz transformation and applying the Khas'minskii formula, under appropriate assumptions on smoothness, ergodicity and integrability. Finally, two examples are present to illustrate our results. The results characterize the growth or decay rates of a class of dynamical systems under the interaction between Hamiltonian structures and non-Gaussian uncertainties.