Do numerical orbits of chaotic dynamical processes represent true orbits?

Stephen M. Hammel*, James A. Yorke, Celso Grebogi

*Corresponding author for this work

Research output: Contribution to journalArticle

171 Citations (Scopus)

Abstract

Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. In an iterated process, if errors double each iterate and numerical calculations have 50-bit (or 15-digit) accuracy, a true orbit through a point can be expected to have no correlation with a numerical orbit after 50 iterates. On the other hand, numerical studies often involve hundreds of thousands of iterates. One may therefore question the validity of such studies. A relevant result in this regard is that of Anosov and Bowen who showed that systems which are uniformly hyperbolic will have the shadowing property: a numerical (or noisy) orbit will stay close to (shadow) a true orbit for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report rigorous results for nonhyperbolic systems: numerical orbits typically can be shadowed by true orbits for long time periods.

Original languageEnglish
Pages (from-to)136-145
Number of pages10
JournalJournal of Complexity
Volume3
Issue number2
DOIs
Publication statusPublished - Jun 1987

Fingerprint

Orbits
Orbit
Iterate
Shadowing Property
Hyperbolicity
Digit
Numerical Calculation
Numerical Study
Tend

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • Analysis
  • Computational Mathematics

Cite this

Do numerical orbits of chaotic dynamical processes represent true orbits? / Hammel, Stephen M.; Yorke, James A.; Grebogi, Celso.

In: Journal of Complexity, Vol. 3, No. 2, 06.1987, p. 136-145.

Research output: Contribution to journalArticle

Hammel, Stephen M. ; Yorke, James A. ; Grebogi, Celso. / Do numerical orbits of chaotic dynamical processes represent true orbits?. In: Journal of Complexity. 1987 ; Vol. 3, No. 2. pp. 136-145.
@article{9de08234ec284aeba590e8988172772e,
title = "Do numerical orbits of chaotic dynamical processes represent true orbits?",
abstract = "Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. In an iterated process, if errors double each iterate and numerical calculations have 50-bit (or 15-digit) accuracy, a true orbit through a point can be expected to have no correlation with a numerical orbit after 50 iterates. On the other hand, numerical studies often involve hundreds of thousands of iterates. One may therefore question the validity of such studies. A relevant result in this regard is that of Anosov and Bowen who showed that systems which are uniformly hyperbolic will have the shadowing property: a numerical (or noisy) orbit will stay close to (shadow) a true orbit for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report rigorous results for nonhyperbolic systems: numerical orbits typically can be shadowed by true orbits for long time periods.",
author = "Hammel, {Stephen M.} and Yorke, {James A.} and Celso Grebogi",
year = "1987",
month = "6",
doi = "10.1016/0885-064X(87)90024-0",
language = "English",
volume = "3",
pages = "136--145",
journal = "Journal of Complexity",
issn = "0885-064X",
publisher = "Academic Press Inc.",
number = "2",

}

TY - JOUR

T1 - Do numerical orbits of chaotic dynamical processes represent true orbits?

AU - Hammel, Stephen M.

AU - Yorke, James A.

AU - Grebogi, Celso

PY - 1987/6

Y1 - 1987/6

N2 - Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. In an iterated process, if errors double each iterate and numerical calculations have 50-bit (or 15-digit) accuracy, a true orbit through a point can be expected to have no correlation with a numerical orbit after 50 iterates. On the other hand, numerical studies often involve hundreds of thousands of iterates. One may therefore question the validity of such studies. A relevant result in this regard is that of Anosov and Bowen who showed that systems which are uniformly hyperbolic will have the shadowing property: a numerical (or noisy) orbit will stay close to (shadow) a true orbit for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report rigorous results for nonhyperbolic systems: numerical orbits typically can be shadowed by true orbits for long time periods.

AB - Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. In an iterated process, if errors double each iterate and numerical calculations have 50-bit (or 15-digit) accuracy, a true orbit through a point can be expected to have no correlation with a numerical orbit after 50 iterates. On the other hand, numerical studies often involve hundreds of thousands of iterates. One may therefore question the validity of such studies. A relevant result in this regard is that of Anosov and Bowen who showed that systems which are uniformly hyperbolic will have the shadowing property: a numerical (or noisy) orbit will stay close to (shadow) a true orbit for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report rigorous results for nonhyperbolic systems: numerical orbits typically can be shadowed by true orbits for long time periods.

UR - http://www.scopus.com/inward/record.url?scp=45949120176&partnerID=8YFLogxK

U2 - 10.1016/0885-064X(87)90024-0

DO - 10.1016/0885-064X(87)90024-0

M3 - Article

VL - 3

SP - 136

EP - 145

JO - Journal of Complexity

JF - Journal of Complexity

SN - 0885-064X

IS - 2

ER -