The Reduction Principle for Systems with Piecewise Constant Argument

Abstract

The theory of integral manifolds founded by H. Poincaré and A. M. Lyapunov [215, 267] became a very powerful instrument for investigating problems of the qualitative theory of differential equations. Over the past several decades, many researchers have been studying the methods of reducing high dimensional problems to low dimensional ones. If we discuss this problem for long-time dynamics of differential equations, we should consider the Reduction Principle [264, 265]. For a brief history of the principle, the reader is referred to the papers [199, 219, 264]. As it is well known that the principle was utilized in the center manifold theory, as well as in the theory of inertial manifolds [68, 118, 155]. On the other hand, it is natural that the exploration of the properties and neighborhoods of manifolds is one of the most interesting problems of the theory of differential equations [59, 68, 74, 153, 180, 248, 268]. One should not be surprised that integral manifolds and the reduction principle are among the major subjects of investigation for specific types of differential and difference equations [39,51,73,75,100,118,149,150,200,249–251,268,294]. The main novelty of this chapter is to extend the principle to differential equations with piecewise constant arguments.