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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Shadowing orbits of ordinary differential equations on invariant submanifolds


Author: Brian A. Coomes
Journal: Trans. Amer. Math. Soc. 349 (1997), 203-216
MSC (1991): Primary 34A50; Secondary 65L70
DOI: https://doi.org/10.1090/S0002-9947-97-01783-2
MathSciNet review: 1390974
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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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Additional Information

Brian A. Coomes
Affiliation: Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124
Email: coomes@math.miami.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01783-2
Keywords: Ordinary differential equations, shadowing, Hamiltonian systems, first integrals, invariant manifolds
Received by editor(s): May 17, 1995
Article copyright: © Copyright 1997 American Mathematical Society