Studies Concerning Applications Of Qualitative Theory On Several Classes Of Filippov Systems And Smooth Differential Systems

Due to the defciency of mathematical analysis tools, the development of theo-ry concerning diferential equations with discontinuous right-hand sides is extreme-ly slow, which is still on the primary stage up to now. Thus, most of theories ondiferential equations with discontinuous right-hand sides are incomplete. To endthis, this thesis frstly flls some gaps of stability theories on non-autonomous gen-eralized homogeneous diferential equations with discontinuous right-hand sides,which mainly discusses the convergence ways of solutions of non-autonomous gen-eralized homogeneous systems with discontinuous right-hand sides and a kind ofdiscontinuous homogeneous perturbed systems. Then, according to characteris-tics of the interaction between preys and predators in our real life, we proposea prey-predator model with discontinuous functional response function and applyqualitative theories of diferential equations with discontinuous right-hand sides tocompletely study this model. In the last two parts of this thesis, by constructingLyapunov functions, we discuss the stability of steady-state of a class of smoothHIV compartment models and a neural network model with discontinuous acti-vation function, respectively. It is valuable to mention that when we study thestability of steady-state in the neural network model with discontinuous activationfunction, we obtain the steady-state is fnite time stable under certain conditions.It is impossible for smooth HIV compartment models to own this style of stability.Specifcally, this thesis is divided into six chapters.In the frst chapter, the development of theories on diferential equations withdiscontinuous right-hand sides and stability are frstly recalled. Subsequently, weintroduce the historical background and status of development concerning dis-continuous biological dynamics, HIV dynamical systems and discontinuous neuralnetwork models, one by one.In the second chapter, some basic mathematical theories which are needed inthis thesis are briefy introduced, mainly including qualitative theories on Filippovsystem, such as the existence and uniqueness of solution, dependence of solutionson initial data and on the right hand side of equation, properties of solutions onthe discontinuous surface in planar Filippov system, topological structure of Σ-singular point. Meanwhile, some stability theories of smooth dynamical systemand Filippov system are also stated here.In the third chapter, we develop stability theories on non-autonomous gen- eralized homogeneous differential equations with discontinuous right-hand sides. By using the property of homogeneous differential equation (scaled solutions are also solutions of original system) and some skills on computation, we obtain con-vergence ways of non-autonomous generalized homogeneous differential equations with discontinuous right-hand sides. We show exponential stability of zero de-gree and l/t type stability for positive degree homogeneous systems, respectively. Meanwhile, we analyze dynamical behaviors of a kind of perturbed system with inner and outer perturbations. The results show that the dynamical behaviors of perturbed systems are similar with that of nominal system and the convergence way of solution is stable under sufficiently small perturbations.In the forth chapter, according to characteristics of the interaction between ratio-dependent preys and predators, we develop a ratio-dependent Filippov prey-predator model. Theories on topological structures of∑-singular points in dif-ferential equations with discontinuous right hand sides are applied to study local structures of all the∑-singular points of our model. In addition,14types of global behaviors are realized for various parameter values. In particular, globally stable pseudo-equilibrium and globally finite time stable cycle are shown to exist in some ranges of parameter values. Moreover, some observations in the experiments and real life can be suitably explained by some dynamical behaviors of our model.In the fifth chapter, we combine the method of constructing Lyapunov function with LaSalle invariant principle to explore the asymptotical stability of steady-state in a kind of seven dimensional and eight dimensional smooth HIV compartment models, respectively. For the two models, we obtain that when the basic repro-duction number is not larger than1(for which the infection equilibrium does not exist), the infection-free equilibrium is globally asymptotically stable; while when the basic reproduction number is larger than1(for which the infection equilibrium exists), the infection equilibrium is globally asymptotically stable.In the sixth chapter, based on the tangency or non-tangency of periodic solu-tion to discontinuous surface, we study finite time stability of periodic solution of a Hopfield neural networks model with discontinuous activation function. When periodic solution is tangent to one of the discontinuous surfaces, it is finite time stable. When periodic solution intersects with all the discontinuous surfaces under some conditions, it is finite time stable by constructing Lyapunov function.