Shadowing deals with the existence of true orbits of dynamical systems near approximate orbits with sufficiently small local errors. Although it has roots in abstract dynamical systems, recent developments have made shadowing into a new effective tool for rigorous computer-assisted analysis of specific dynamical systems, especially chaotic ones. For instance, using shadowing it is possible to prove the existence of various unstable periodic orbits, transversal heteroclinic or homoclinic orbits of arguably the most prominent chaotic system - the Lorenz Equations. In this paper we review the current state of the theory and applications of shadowing for ordinary differential equations, with particular emphasis on our own work.
|Original language||English (US)|
|Number of pages||25|
|Journal||Rendiconti del Seminario Matematico|
|State||Published - 2007|
ASJC Scopus subject areas