Shadowing orbits of ordinary differential equations

Brian A. Coomes; Hüseyin Koçak; Kenneth J. Palmer

doi:10.1016/0377-0427(94)90347-6

Shadowing orbits of ordinary differential equations

Brian A. Coomes, Hüseyin Koçak, Kenneth J. Palmer

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then 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 pseudo orbit in terms of computable quantities, is proved. The use of this theorem in numerical computations of orbits is outlined.

Original language	English (US)
Pages (from-to)	35-43
Number of pages	9
Journal	Journal of Computational and Applied Mathematics
Volume	52
Issue number	1-3
DOIs	https://doi.org/10.1016/0377-0427(94)90347-6
State	Published - Jul 20 1994

Keywords

Chaos
Numerical orbits
Ordinary differential equations
Shadowing

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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title = "Shadowing orbits of ordinary differential equations",

abstract = "A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then 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 pseudo orbit in terms of computable quantities, is proved. The use of this theorem in numerical computations of orbits is outlined.",

keywords = "Chaos, Numerical orbits, Ordinary differential equations, Shadowing",

author = "Coomes, {Brian A.} and H{\"u}seyin Ko{\c c}ak and Palmer, {Kenneth J.}",

note = "Funding Information: * Corresponding author. {\textquoteright} Supported in part by NSF. * Supported in part by DOE (Office of Scientific Computing).",

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T1 - Shadowing orbits of ordinary differential equations

AU - Coomes, Brian A.

AU - Koçak, Hüseyin

AU - Palmer, Kenneth J.

N1 - Funding Information: * Corresponding author. ’ Supported in part by NSF. * Supported in part by DOE (Office of Scientific Computing).

PY - 1994/7/20

Y1 - 1994/7/20

N2 - A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then 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 pseudo orbit in terms of computable quantities, is proved. The use of this theorem in numerical computations of orbits is outlined.

AB - A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then 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 pseudo orbit in terms of computable quantities, is proved. The use of this theorem in numerical computations of orbits is outlined.

KW - Chaos

KW - Numerical orbits

KW - Ordinary differential equations

KW - Shadowing

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JF - Journal of Computational and Applied Mathematics

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