Dynamics Of Stochastic Hamiltonian Systems With Non-Gaussian Noise

Certain nonlinear systems have “geometric” structures,such as the Hamiltonian structure.Hamiltonian systems of ordinary differential equations widely appear in celestial mechanics,statistical mechanics,geophysics,and chemical physics.Hamiltonian systems have many well-known properties.For example,the flows of these systems possess the property of phase-volume preservation.However,these systems in engineering and so on are inevitably affected by random fluctuations(noise).This leads to some interesting dynamical phenomena.In present thesis,we focus on the dynamical behaviors of stochastic Hamiltonian systems driven by non-Gaussian L(?)vy noise,including symplecticity,Hamilton's principle,averaging principle and Lyapunov exponents.This thesis is composed of the following parts.In Chapter 1,we introduce the research background,current research status,and outline the main results of this thesis.Chapter 2 is devoted to the review of relevant definitions and important theorems of deterministic Hamiltonian systems,L(?)vy motions and stochastic analysis.In Chapter 3,we consider a class of stochastic Hamiltonian systems driven by nonGaussian L(?)vy noise in the Marcus form.After presenting the existence and uniqueness of solutions for such sysems,we show that the phase flow of this stochastic system preserves the symplectic structure,that is,these systems satisfy symplecticity.By considering a stochastic Hamiltonian system with L(?)vy noise as a special nonconservative system,we propose a stochastic version of Hamilton's principle.In Chapter 4,we investigate the effective behaviour of a small transversal perturbation to a completely integrable stochastic Hamiltonian system with L(?)vy noise,and establish an averaging principle.We first study the invariant manifold and invariant measure for the unperturbed system.For the perturbed system,by an action-angle coordinate,we find that the fast rotation is a diffusion on the invariant torus and the slow motion is governed by the transversal component.After discussing the ergodic behavior and some technical issues,we get the information on the order of which the first integrals for the perturbed system change over a time interval,and then show that the action component of solution converges to the solution of a stochastic differential equation,when the scale parameter goes to zero.Furthermore,we obtain the estimation for the rate of this convergence.Finally,we present an example to illustrate these results.In Chapter 5,we focus on estimating the(top)Lyapunov exponent for a class of Hamiltonian systems under small non-Gaussian L(?)vy 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,under appropriate assumptions on smoothness,ergodicity and integrability.This characterizes the growth or decay rates of a class of dynamical systems under the interaction between Hamiltonian structures and non-Gaussian uncertainties.In Chapter 6,we investigate contact Hamiltonian systems which are “odd-dimensional cousins” of symplectic Hamiltonian systems.We devise a stochastic version of contact Hamiltonian systems,and show that their phase flows of these systems preserve contact structures.Moreover,we provide a sufficient condition under which these stochastic contact Hamiltonian systems are completely integrable.This establishes an appropriate framework for investigating stochastic contact Hamiltonian systems.Note that,even for Gaussian case,there is few research work on stochastic contact Hamiltonian systems.In this chapter,we focus on a system driven by Brownian noise and claim that our results could be extended to non-Gaussian case.In Chapter 7,a summary is presented,and the follow-up research questions are discussed as well.