The existence of a true orbit near a numerically computed approximate orbit -shadowing -of autonomous system of ordinary differential equations is investigated. A general shadowing theorem for finite time, which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the approximate orbit in terms of computable quantities, is proved. The practical use and the effectiveness of this theorem is demonstrated in the numerical computations of chaotic orbits of the Lorenz equations. Mathematics Subject Classification (1991): 65L05

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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 submani-fold 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.

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