Practical Stability And Numerical Computation Of Filippov-type Ordinary Differential Equations And Stochastic Differential Equations

In this thesis, we mainly investigate practical stability for Filippov-type ordinary differential equations and stochastic differential equations with discontinuous coefficients,and the numerical computation for stochastic differential equations with discontinuous coefficients.The theory of practical stability is one branch of the modern stability theory, which is applied in impulsive systems, hybrid systems and descriptor systems etc.. The concept of practical stability is usually defined in terms of neighborhoods of a point [49] and is applied to the local stability property. In general, it is not enough to study the stability near a trivial solution. Consequently, it is useful to extend the practical stability notions relative to arbitrary sets or more general, arbitrary tubes. In [49, 86], the practical stability is generalized to deal with two arbitrary setsΩ₁ (?)Ω₂,which are specified for the initial state and the entire system state, respectively. Obviously, the two setsΩ₁ andΩ₂ do not have to include a trivial solution and they can be specified flexibly in real applications. Then, this generalized practical stability (GP-stability for short) requires that if the initial state is inΩ₁,then the system state should always stay inΩ₁.The GP-stability is a significant extension of practical stability.In the first part of the thesis, we study the GP-stability for Filippov-type ordinary differential equations. By using the Lyapunov-like functions and the properties of the solution to Filippov-type ordinary differential equations, criteria are established for determining the GP-stability and the totally GP-stability. We also establish the criteriafor determining the GP-stability and the totally GP-stability under small perturbation. We apply our theories to the brake model of a bike and 1-DOF model. For the brake model of a bike, Zou[90] obtained the piecewise smooth systems containing the parameter. He obtained the existence of bifurcating periodic orbit via generalized Hopf bifurcation, and studied the stability of this bifurcating periodic orbit via the numerical computation. We study the stability of this bifurcating periodic orbit in theory. We prove that the bifurcating periodic orbit is uniformly GP-stability and the totally GP-stability under small perturbation. We present some numerical computation to illustrate our theories.Another application for Filippov-type ordinary differential equations is 1-DOF model[9]. 1-DOF model is the model with dry friction which expose stick-slip vibrations.This model can be abstracted as Filippov-type ordinary differential equations. Kunze[45] proved the existence of periodic solutions for the equations. We prove the periodic solutions are strong GP-stability, and present some numerical computation to illustrate our theories.With the development of modern mathematics, the theory of stochastic differential equation becomes even more important. It can be used to describe many sophisticated dynamical systems in physical, biological, technical, financial, social sciences etc. [69,70,74]. Feng [12] applied the basic comparison principle to stochastic systems [13, 14] and established criteria for various types of practical stability in the pth mean of nonlinear stochastic systems. In real life, the coefficients of the stochastic differential equation is always discontinuous. For example, Continuous-Time Threshold Autoregressive Moving Average(CTARMA) model is a useful model. The drift is piecewise linear and the diffusion term is picecwise constant.In the second part of the thesis, we study the practical stability in the pth mean of the following stochastic differential equations with discontinuous coefficients:where b(·) areσ(·) are locally bounded Borel measurable functions. Under appropriate conditions, the existence of solutions for equation (1) is proved in the literature [13]. By using Lyapunov-like function, we establish the criteria of determining the practical stability in the pth mean of stochastic differential equations with discontinuous coefficients.Assuming the existence and uniqueness of solution, we study the uniformly practical stability in the pth mean of the following stochastic differential equations with discontinuous coefficients:We also present some numerical computation to illustrate our theories.The systematic study on the numerical methods for the stochastic differential equations with continuous coefficients is collected in the book [43]. The numerical methods for the stochastic differential equations with discontinuous coefficient are also paid attention to [13, 73, 82, 30].In the third part of the thesis, we study the numerical method for the following stochastic differential equations with discontinuous coefficients:where b(·) area are locally bounded Borel measurable functions. Under appropriate conditions, the Euler scheme is weakly convergence[13].We apply Heun scheme to solve equation (2). We prove that Heun scheme is weakly convergence under some conditions. The convergence rates of Euler scheme and Heun scheme for equation (2) are not confirmed until now. Under some continuous conditions, the weak convergence rate of Euler scheme for the stochastic differential equation is 1, while the weak convergence rate of Heun scheme is 2. We guess the Heun scheme is convergence faster than Euler scheme. We present some numerical computation to illustrate this guess.