Shadowing orbits of ordinary differential equations on invariant submanifolds

Coomes, Brian

doi:10.1090/S0002-9947-97-01783-2

Shadowing orbits of ordinary differential equations on invariant submanifolds
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Trans. Amer. Math. Soc. 349 (1997), 203-216 Request permission

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 $\mathbb {R}^{m}$ 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.

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