Quantifying Non-gaussian Stochastic Dynamical Systems

In engineering and applied science,the random factors such as environment vari-abilities,initial conditions and boundary conditions,cause the uncertainty of the mathe-matical models in complex dynamical systems.I have consider two types of stochastic noises in modeling stochastic dynamics.The first one is Gaussian white noise,and the other one is non-Gaussian white noise.Gaussian white noise is a type of simple and common noise,which is often modeled by Brownian motions.It arises in a wide range of phenomenon in uncertainty complex systems.However,various complex sys-tems evolve in random manners with non-Gaussian noise,such as the climate change,foraging movement patterns for marine predators,the production of mRNA and pro-tein in gene regulatory system.Non-Gaussian noise is appropriately modeled by the Levy motions.Meanwhile,the ?-stable Levy motion is a special but important type of Levy motion defined by the stable Levy random variables.The main tasks in this thesis are to quantify statistic properties and stochastic dynamical behaviors with ?-stable Levy motions by deterministic quantities,including the mean first exit time,first escape probability,stochastic basin of attraction and probability density functions.This thesis is organized as follows:In Chapter 1,I have presented and revised some of basic conceptions and proper-ties of the stochastic analysis,Brownian motion,Levy process and stochastic dynami-cal systems.Chapter 2 is about a stochastic differential equation model for a single genetic regulatory system.I have examined the dynamical effects of noisy fluctuations from the synthesis reaction,on the evolution of the transcription factor activator in terms of its concentration.The fluctuations can be modeled by Brownian motion and ?-stable Levy motion,respectively.Two deterministic quantities,the mean first exit time(MFET)and the first escape probability(FEP),are used to investigate the transition from the low to high concentration states.A shorter MFET or higher FEP in the low concentration region is in favour of transition.I have observed that higher noise intensities and larger jumps of the Levy motion reduce the MFET.The Levy motion activates a transition from the low concentration region to the non-adjacent high concentration region,while Brownian motion can not induce this phenomenon.There are optimal proportions of Gaussian and non-Gaussian noises,which maximise the quantities MFET and FEP for each concentration,when the total sum of noise intensities are kept constant.In Chapter 3,I have introduced a new concept called stochastic basin of attraction for stochastic dynamical systems.It will be used to quantify the basin stability of s-tochastic dynamics and the characteristic of metastability.A concrete approach based on the MFET u(x),and the FEP p(x)is utilized to determine the stochastic basin of attraction.Two type of definitions for stochastic basin of attraction are given.Stochas-tic basin of attraction is the set of all initial concentrations whose trajectories have a high probability to return to the set where the stochastic solutions may not escape with a large probability or remain in a long(but finite)time.Considering the modeling of transcription factor activator monomer concentration system,I have observed the nu-merical result and have showed that the size of stochastic basin of attraction effects on the criterion of MFET and FEP,and the Levy process index a.In Chapter 4,I have considered the probability density evolution by Marcus s-tochastic differential equations,which often are appropriate models for stochastic dy-namical systems driven by non-Gaussian Levy processes and have wide applications in engineering and physical sciences.The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process.Explic-it formula for the Fokker-Planck equation,the governing equation for the probability density,is well-known when the SDE is driven by a Brownian motion.In this chapter,I have addressed the open question of finding the Fokker-Plank equation for Marcus SDEs in driven by multi-dimensional non-Gaussian Levy processes and with more general noise coefficients.The equations are given in a simple form that facilitates theoretical analysis and numerical computation.Several examples are presented to il-lustrate how the theoretical results can be applied to obtain Fokker-Planck equations for Marcus SDEs driven by Levy processes.In the final chapter,I have presented the result in this thesis and proposed some future research tops.