Shadowing orbits of ordinary differential equations on invariant submanifolds

Abstract

Abstract. A finite time shadowing theorem for autonomous ordinary differential equations is presented. Under consideration is the case were there exists a twice continuously differentiable function g mapping phase space into R m with the property that for a particular regular value c of g the submanifold g −1 (c) is invariant under the flow. The main theorem gives a condition which implies that an approximate solution lying close to g −1 (c) is uniformly close to a true solution lying in g −1 (c). Applications of this theorem to computer generated approximate orbits are discussed.

Chaos on Hyperspace

“…On the other hand, the approach of shadowing orbits methodology implies also very fruitful contribution to the stable numerical methods, see, e.g. [Pilyugin, 1999;Coomes, 1997].…”