Asymptotic Behaviors Of Discontinuous Dynamical Systems And Its Applications

This paper is concerned with asymptotic behaviors for a class of discontinuous dynamicalsystems—– impulsive dynamical systems, and its applications to neural networks, populationsystems and chaotic systems.In Chapter 1, we discuss the asymptotic behavior for impulsive functional (partial) differen-tial equations. Firstly, we obtain some criteria to determine the attracting set and attracting basinby some methods of modern analysis. Then, the existence and exponential stability of impulsivenonlinear functional differential equations by using the impulsive delay inequalities and introducea Banach space PCt. Lastly, we discuss the invariant and attracting sets of a class of impulsivepartial functional differential equation.In Chapter 2, by introducing M?cone and developing the impulsive differential inequalities,we obtain some sufficient conditions of the stability for some neural networks with delays andimpulses, including Hopfield type neural networks, Cohen-Grossberg neural networks and theneural networks described by measure differential equations.In Chapter 3, we frst discuss the existence and global asymptotical stability for Lotka-Volterra competitive systems with feedback controls, impulsive effects and infinite distributeddelays by applying the Mawhin's continuous theorem and the Lyapunov functional approach.Also, by the impulsive type Barbalet lemma and impulsive compared result, we investigate thepersistence , the existence, uniqueness and global attractivity of Holling-III type predator-preysystems with impulsive effects.In Chapter 4, we discuss impulsive control for the stabilization and synchronization of somechaotic systems including delayed chaotic systems, large-scale chaos and spatiotemporal chaoticsystems.