Shadowing orbits of ordinary differential equations

Brian A Coomes, Huseyin Kocak, Kenneth J. Palmer

Research output: Contribution to journalArticle

15 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)35-43
Number of pages9
JournalJournal of Computational and Applied Mathematics
Volume52
Issue number1-3
DOIs
StatePublished - Jul 20 1994

Fingerprint

Shadowing
Ordinary differential equations
Ordinary differential equation
Orbits
Orbit
Autonomous Systems
System of Ordinary Differential Equations
Theorem
Numerical Computation
Error Bounds
Approximate Solution

Keywords

  • Chaos
  • Numerical orbits
  • Ordinary differential equations
  • Shadowing

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

Shadowing orbits of ordinary differential equations. / Coomes, Brian A; Kocak, Huseyin; Palmer, Kenneth J.

In: Journal of Computational and Applied Mathematics, Vol. 52, No. 1-3, 20.07.1994, p. 35-43.

Research output: Contribution to journalArticle

@article{92130e27a00a4d3dad2f1b098c2f1aac,
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 Huseyin Kocak and Palmer, {Kenneth J.}",
year = "1994",
month = "7",
day = "20",
doi = "10.1016/0377-0427(94)90347-6",
language = "English (US)",
volume = "52",
pages = "35--43",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",
number = "1-3",

}

TY - JOUR

T1 - Shadowing orbits of ordinary differential equations

AU - Coomes, Brian A

AU - Kocak, Huseyin

AU - Palmer, Kenneth J.

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

UR - http://www.scopus.com/inward/record.url?scp=0028479087&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028479087&partnerID=8YFLogxK

U2 - 10.1016/0377-0427(94)90347-6

DO - 10.1016/0377-0427(94)90347-6

M3 - Article

AN - SCOPUS:0028479087

VL - 52

SP - 35

EP - 43

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-3

ER -