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