Shadowing orbits of ordinary differential equations on invariant submanifolds

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 uniformlyclose to a true solution lying in g^-1(c). Applications of this theorem to computer generated approximate orbits are discussed.