New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems

Rafail V. Abramov, Andrew J. Majda

Research output: Contribution to journalArticle

Abstract

We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation-dissipation theorem. Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation-dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.

Original languageEnglish (US)
Pages (from-to)303-341
Number of pages39
JournalJournal of Nonlinear Science
Volume18
Issue number3
DOIs
StatePublished - Jun 2008

Fingerprint

Dissipative Dynamical System
Chaos theory
Fluctuation-dissipation Theorem
Dynamical systems
Fluctuations
Forcing
Linear Response
Approximation
Nonlinear Function
Fractals
Response Time
Nonlinear systems
Dissipation
Chaos
Chaotic Dynamical Systems
Gaussian Approximation
Prediction
Dissipative Systems
Nonlinear Response
Rational Approximation

ASJC Scopus subject areas

  • Applied Mathematics
  • Modeling and Simulation
  • Engineering(all)

Cite this

New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems. / Abramov, Rafail V.; Majda, Andrew J.

In: Journal of Nonlinear Science, Vol. 18, No. 3, 06.2008, p. 303-341.

Research output: Contribution to journalArticle

@article{3a255bc9a0994096b9c1e703afc1aa13,
title = "New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems",
abstract = "We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation-dissipation theorem. Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation-dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.",
author = "Abramov, {Rafail V.} and Majda, {Andrew J.}",
year = "2008",
month = "6",
doi = "10.1007/s00332-007-9011-9",
language = "English (US)",
volume = "18",
pages = "303--341",
journal = "Journal of Nonlinear Science",
issn = "0938-8974",
publisher = "Springer New York",
number = "3",

}

TY - JOUR

T1 - New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems

AU - Abramov, Rafail V.

AU - Majda, Andrew J.

PY - 2008/6

Y1 - 2008/6

N2 - We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation-dissipation theorem. Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation-dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.

AB - We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation-dissipation theorem. Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai-Ruelle-Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation-dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.

UR - http://www.scopus.com/inward/record.url?scp=84867954551&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867954551&partnerID=8YFLogxK

U2 - 10.1007/s00332-007-9011-9

DO - 10.1007/s00332-007-9011-9

M3 - Article

AN - SCOPUS:84867954551

VL - 18

SP - 303

EP - 341

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 3

ER -