Shadowing orbits of ordinary differential equations on invariant submanifolds

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 g mapping phase space into  R m \mathbb {R}^{m} with the property that for a particular regular value  c \boldsymbol c of  g g the submanifold  g − 1 ( c ) g^{-1}(\boldsymbol c) is invariant under the flow. The main theorem gives a condition which implies that an approximate solution lying close to  g − 1 ( c ) g^{-1}(\boldsymbol c) is uniformly close to a true solution lying in  g − 1 ( c ) g^{-1}(\boldsymbol c) . Applications of this theorem to computer generated approximate orbits are discussed.