Shadowing in perturbation theory

Abstract

An approximate solution of a differential equation is said to shadow a true solution if it remains close to the true solution for all time, although it may not coincide exactly with the true solution at any time. Both the regular perturbation method and the method of averaging are usually viewed as methods of solving initial value problems; therefore the approximate solution produced by these methods is expected to coincide with the true solution at the initial time. The resulting approximate solutions do not shadow the true solution, but break down after some time has passed. It is shown that under certain conditions of hyperbolicity and transversality, both methods lead to solutions which satisfy shadowing to any specified order in the small parameter, provided that the initial value requirement is dropped.