Stable Invariant Manifolds Of Nonlinear Impulsive Differential Equations

An impulsive differential equation characterizes a smooth evolution of a dynam-ics that at certain times changes instantaneously and reflect the variety regulation of objective things. There are many applications of these equations to mechanical and natural phenomena involving abrupt changes, including in ecology, medicine, physics, aerospace and control engineering. The existence of the invariant manifolds plays an important role in the study of qualitative and stability of the solutions of dynamical systems and provides a geometric description to understand the global structure of dynamical systems. Therefore, the paper focuses on the existence of stable invariant manifolds for nonlinear impulsive differential equations.In this paper, we systematically explore the problem of the existence of stable invariant manifolsd of nonlinear impulsive differential equations. Firstly, we intro-duce the concept of the nonuniform (h, k,?, v)-dichotomy for the linear impulsive differential equations. With the help of nonuniform (h, k,?,v)-dichotomies, we es-tablish the existence of Lipschitz stable invariant manifolds of nonlinear impulsive differential equations, and show that the stable invariant manifolds are Lipschitz continuous for the initial values, the parameters and the right-side functions. Sec-ondly, we establish the existence of the stable manifold with continuous differentiable property for nonlinear impulsive differential equations. Finally, we summarize the main results of this paper and point out the problems to be studied in the further.