Theory Of Differential Equations With Discontinuous Righthand Sides And Its Applications

In this thesis, we first extend and generalize the existing theory of differential equa-tions with discontinuous righthand sides (discontinuous differential equations) by combinginclusion theory, perturbation theory of differential equations and differential inequalitytechniques. The differential equations and Lyapunov functions we considered are moregeneral, and the obtained results provide a powerful tool for the applications of discontin-uous differential equations in many areas. Then, by using the theory of discontinuous dif-ferential equations, we discuss the dynamical behaviors of neural networks with discontin-uous neuron activations, fishing model with discontinuous harvesting policy and epidemicmodel with discontinuous treatments. These results have important practical significanceand theoretical guidance for designing of stable neural networks, effective and sustainabledevelopment of fishery resources, prevention and control of the infectious diseases.As the introduction, in Chapter One, a review on the development of the theory ofdiscontinuous differential equations is presented, the background and history of dynamicsfor neural networks, fishing model, the infectious diseases are also briey addressed. Themotivations and outline of this work are also given in this chapter.In Chapter Two, we consider the discontinuous ordinary differential equations anddelayed differential, and the solutions in Filippov sense of these two kinds of equations.Based on Filippov differential inclusion and a new chain rule for differentiating regularfunctions along Filippov solution trajectories, different kinds of stability and convergenceresults are presented. Moreover, we investigate the stability and convergence for the cor-responding perturbation system of the discontinuous ordinary differential equations andsome new criteria are addressed.In Chapter Three, we investigate the dynamics of three class of neural networks withdiscontinuous neuron activations, which are autonomous, periodic and more general non-autonomous neural networks. Our methods are based on the generalized Lyapunov theory,matrix theory, fixed theory and some inequalities techniques. Firstly, without assuming theneuron activations are bounded, some global robust stability criteria for a class of delayedneuron networks with discontinuous neuron activations are addressed. Next, droppingthe assumption that the neuron activations are bounded and monotonic, we study the ex-istence and global asymptotical stability for the periodic solution of a class of periodicneural networks with discontinuous neuron activations. Finally, we consider two kinds of non-autonomous neural networks with non-periodic coefficients and discontinuous neuronactivations, and obtain some new results on global output convergence, global state con-vergence and global convergence in finite time. We also give many numerical examples toshow the effectiveness of our results in this chapter.Then, in Chapter Four, in virtue of generalized Lyapunov theory and Variable Struc-ture Control Theory, we make a investigation of a fishing model with discontinuous har-vesting policy. some sufficient conditions is given for assuring local stability, global stabil-ity and the appearance of slide mode. From the view point of economics and sustainabledevelopment, we give the rational explanations of the obtained results.Finally, in Chapter Five, a class of epidemic model with discontinuous treatmentsis considered. By using generalized Lyapunov theory and Lasalle invariance principle,we discuss the local stability, global stability and global convergence in finite time of themodel. The discontinuous treatment strategies generalize the treatment strategies consid-ered in previous literature. By the biological explanations, we can understand the dis-continuous treatment strategies better, and realize the effective control and elimination ofinfectious diseases.