Shadowing Theorems for Hyperbolic Sets of Differential Equations

Abstract

For differential equations, it is not immediately clear how the shadowing theorem should be formulated and, in particular, how pseudo orbits should be defined. We consider two possibilities: first a discrete pseudo orbit such as would be obtained by numerically computing the solutions of a differential equation and then a continuous pseudo orbit which is needed for theoretical purposes. Shadowing theorems are proved for both kinds of pseudo orbits. We also study both discrete and continuous versions of expansivity. Then shadowing is used to show that there is a topological conjugacy between the flow on an isolated hyperbolic set and the flow on the nearby hyperbolic set for a perturbed flow. Also we use shadowing to show that isolated hyperbolic sets have the asymptotic phase property.