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Slow Manifolds for Multiscale Stochastic Dynamical Systems-Theory, Approximation and Application

Author: RenJian
Tutor: DuanJinQiao
School: Huazhong University of Science and Technology
Course: Probability Theory and Mathematical Statistics
Keywords: Multi-time scale systems Stochastic ordinary differential equations Random dynamical systems Dimension reduction Approximation Random invariantmanifolds Slow manifolds Invariant foliation Exponential decay/attracting property Stochastic optimization
CLC: O19
Type: PhD thesis
Year: 2013
Downloads: 59
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Complex dynamical systems in science and engineering often involve multiple time scales, such as slow and fast time scales, as well as uncertainty caused by noisy fluctuations. For example, aerosol and pollutant particles, occur in various natural contexts (e.g., in atmosphere and ocean coasts) and engineering systems (e.g. spray droplets), are described by coupled stochastic dynamical systems. Some particles move faster while others move slower, and they are usually subject to random influ-ences, due to molecular diffusion, environmental fluctuations, or other small scale mechanisms that are not explicitly modeled. The coupled stochastic dynamical sys-tems are stochastic ordinary differential slow-fast systems, stochastic partial differ-ential slow-fast systems and microscopic-macroscopic stochastic slow-fast systems. There has been great interest for the research in this area.In the study of the properties of a flow or semiflow near an invariant set, invariant manifolds and invariant foliations are fundamental tools. Here we study the invariant manifold (slow manifold) and invariant foliation of two-time scale slow-fast stochas-tic dynamical systems. Firstly, we consider the existence of slow manifold and its approximation for slow-fast stochastic ordinary differential equations, together with invariant foliation and its expansion. Decay of the orbits for stochastic differential equations to its slow manifold have also been given. The reduced system is given by restricting the system on the slow manifold. Then, we discuss two applications based on random slow manifold reduced system:settling of the aerosol particles disturbed by noise and a problem of parameter estimation.In Chapter1, we review some basic concepts and definitions about stochastic process and random dynamical systems.In Chapter2, stochastic dynamical system with noise is explored. The system is described by a slow-fast system with noise in the fast equation. Different from the random graph transformation method, here we give the proof of the existence of a random slow manifold by the Lyapunov-Perron method. We then find a method to approximate the slow manifold. We use invariant foliation to find a invariant set witch consists of initial points, the initial points on a fiber/leaf decay with exponential rate. Moreover, we find a method to approximate fibers or leaves of the invariant foliation. At last, an example has been given to demonstrate the slow manifold and invariant foliation.In Chapter3, we consider the settling of inertial/aerosol particles under uncer-tainty, by the help of a random slow manifold. The motion of a particle is described by a stochastic slow-fast system consists of position parameters and velocity parame-ters of the particles. By slow manifold, we can kill the velocity parameters and get a reduced system with position parameters only. By numerical simulation, we consider the first exit time and exit probability of the particles. And we get some results about the motion of the particles, which are different from the deterministic case even with inertial.In Chapter4, We devise a new parameter estimation method by using random slow manifolds. Different from the method based on the original slow-fast system, here we only need the slow component observable. By using the available observa-tions on the slow component, the system parameters are estimated by working on the slow system. This offers a benefit of dimension reduction in quantifying parameters in stochastic dynamical systems. An example is presented to illustrate this method, and verify that the parameter estimator based on the lower dimensional reduced slow system is a good approximation of the parameter estimator from the original slow-fast stochastic dynamical system.In the final chapter, a summary of this thesis is presented and some questions that need to be improved and perfected in the future study are raised.

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