This work is to investigate the (top) Lyapunov exponent for a class of Hamiltonian systems under small non-Gaussian Lévy-type noise with bounded jumps. In a suitable moving frame, the linearization 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 presented to illustrate our results.
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