Dynamics Of Non-Gaussian Stochastic Dynamical Systems

Most complex systems in the real world are inevitably affected by random fluctuations.These disturbances promote many interesting phenomena in dynamical systems,such as transition between metastable states,stochastic resonance,and a narrow escape problem.It is of great significance to understand and describe the dynamical behaviors of stochastic dynamical systems.Mathematically,stochastic differential equations are suitable models for describing stochastic systems,and their construction need to consider the type of noise.Gaussian white noise is a type of simple and common noise,but non-Gaussian Lévy noise is more universal.For unpredictable or burst-like behaviors in climate change,earthquakes or stock market crashes,Lévy noise is a more appropriate descriptor.The present thesis focuses on the two topics related to the dynamical behaviors of stochastic differential equations driven by the non-Gaussian Lévy process.The first topic is devoted to investigating the most probable path connecting metastable states,and deriving the Onsager-Machlup function in the non-Gaussian sense,which is regarded as the Lagrangian.The second one is concerned with stable and unstable invariant foliations as well as its relationship to invariant manifolds.The results in both topics indicate some certain stable or unstable property of stochastic dynamic systems,but respectively they have their own emphasis.The former result is often used to study the noise-induced transition phenomenon,while the latter one is used to describe the invariant geometric structures.In each topic,some illustrative examples for the results are given.As a further important application of random invariant manifolds,we develop a parameter estimation method for a fast-slow stochastic system with non-Gaussian Lévy noise.This thesis is organized as follows:In Chapter 1,we introduce the research background,current research status,and main research contents of this thesis.In Chapter 2,we review the concepts and theorems related to Brownian motion and Lévy process.At the same time,we present the relevant conclusions of the Marcus stochastic differential equation.Chapter 3 is dedicated to deriving the Onsager-Machlup functional for a class of stochastic dynamical systems under(non-Gaussian)Lévy noise as well as(Gaussian)Brownian noise,and examining the corresponding most probable paths.Note that this OnsagerMachlup theory for the most probable paths does not require noise to be small.Thus it is different from the most probable paths based on large deviation principles,which hold for small noise.In this chapter,the Onsager-Machlup functional is obtained by applying the Girsanov transformation for measures induced by solution processes.Moreover,the new functional is quite generally so that it can recover the results in the special case of diffusion processes.At the same time,via variational principle,we give the most probable path connecting metastable states for these stochastic systems.In other words,we present the equation of motion for the most probable path of a class of stochastic systems and discuss its solution.Finally,some concrete examples are tested to illustrate how the theory is applied to the study of the most probable dynamics.These numerical experiments show the effect of non-Gaussian noise parameters on the most probable transition paths,thereby confirming the non-trivial effect of pure Lévy noise in stochastic dynamics.In Chapter 4,we mainly focus on the invariant geometric structures in stochastic dynamical systems driven by a non-Gaussian Lévy process.We consider a class of Marcus stochastic differential equations,which are widely used in engineering,physics and other fields.Mathematically,it also has the advantage of stochastic calculus,that is,it satisfies the Newton-Leibniz type chain rule.Motivated by the ideas from deterministic dynamical systems,we establish the existence of stable and unstable foliations for this kind of stochastic systems.This is done by using Lyapunov-Perron method and random transformation based on the generalized Ornstein-Uhlenbeck process.Moreover,we also examine the geometric structure of the invariant foliation and further illustrate a link with invariant manifolds.These findings provide a geometric visualization for the state space of the dynamical system.Finally,we try to give an example to verify the validity of this theory.The invariant geometric structure theory established in this chapter can provide a certain basis for subsequent applied research.In Chapter 5,we investigate the parameter estimation for a fast-slow stochastic system with non-Gaussian Lévy noise.We consider a type of reduction method for these systems through the random slow manifolds.By focusing on the reduced slow system on the random slow manifold,an unknown system parameter in the drift term of such systems is estimated.And the accuracy for this estimation is quantified by p-moment with p ?(1,?).Instead of solving original stochastic systems,this method use the observations only available for slow components,and thus it provides an advantage in computational complexity and cost.Furthermore,we corroborate this method numerically,i.e.,the parameter estimator based on the reduced slow system is a good approximation for the true parameter value of the original system.Finally,we give some discussions and comments in a more biological context.In Chapter 6,we summarize the current research work,and point out the follow-up research questions and directions.