Low-dimensional chaotic dynamics versus intrinsic stochastic noise: A paradigm model

A. Majda, I. Timofeyev

Research output: Contribution to journalArticle

Abstract

Several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low-dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables. These models are built by coupling a four-dimensional ODE with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables. A systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four-dimensional reduced stochastic equations for the slow variables. Due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system. Furthermore, it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath, the detailed structure of the stochastic terms derived through the mode-elimination procedure is essential for reproducing the statistical behavior of the slow dynamics.

Original languageEnglish (US)
Pages (from-to)339-368
Number of pages30
JournalPhysica D: Nonlinear Phenomena
Volume199
Issue number3-4
DOIs
StatePublished - Dec 15 2004

Fingerprint

Chaotic Dynamics
Paradigm
baths
Heteroclinic Cycle
Chaos theory
Heat Bath
Model
Stochasticity
Reduced Model
Weak Coupling
Nonlinear Dynamics
Stochastic Equations
Elimination
chaos
elimination
Chaos
degrees of freedom
Degree of freedom
energy transfer
prototypes

Keywords

  • Heteroclinic orbit
  • Mode reduction
  • Non-Gaussian statistics
  • Stochastic modeling

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Low-dimensional chaotic dynamics versus intrinsic stochastic noise : A paradigm model. / Majda, A.; Timofeyev, I.

In: Physica D: Nonlinear Phenomena, Vol. 199, No. 3-4, 15.12.2004, p. 339-368.

Research output: Contribution to journalArticle

@article{5d5fc72ed9964271aec2d730b2d69857,
title = "Low-dimensional chaotic dynamics versus intrinsic stochastic noise: A paradigm model",
abstract = "Several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low-dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables. These models are built by coupling a four-dimensional ODE with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables. A systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four-dimensional reduced stochastic equations for the slow variables. Due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system. Furthermore, it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath, the detailed structure of the stochastic terms derived through the mode-elimination procedure is essential for reproducing the statistical behavior of the slow dynamics.",
keywords = "Heteroclinic orbit, Mode reduction, Non-Gaussian statistics, Stochastic modeling",
author = "A. Majda and I. Timofeyev",
year = "2004",
month = "12",
day = "15",
doi = "10.1016/j.physd.2004.05.012",
language = "English (US)",
volume = "199",
pages = "339--368",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "3-4",

}

TY - JOUR

T1 - Low-dimensional chaotic dynamics versus intrinsic stochastic noise

T2 - A paradigm model

AU - Majda, A.

AU - Timofeyev, I.

PY - 2004/12/15

Y1 - 2004/12/15

N2 - Several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low-dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables. These models are built by coupling a four-dimensional ODE with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables. A systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four-dimensional reduced stochastic equations for the slow variables. Due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system. Furthermore, it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath, the detailed structure of the stochastic terms derived through the mode-elimination procedure is essential for reproducing the statistical behavior of the slow dynamics.

AB - Several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low-dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables. These models are built by coupling a four-dimensional ODE with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables. A systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four-dimensional reduced stochastic equations for the slow variables. Due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system. Furthermore, it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath, the detailed structure of the stochastic terms derived through the mode-elimination procedure is essential for reproducing the statistical behavior of the slow dynamics.

KW - Heteroclinic orbit

KW - Mode reduction

KW - Non-Gaussian statistics

KW - Stochastic modeling

UR - http://www.scopus.com/inward/record.url?scp=10644246719&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10644246719&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2004.05.012

DO - 10.1016/j.physd.2004.05.012

M3 - Article

AN - SCOPUS:10644246719

VL - 199

SP - 339

EP - 368

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -