Ergodic theory of chaos and strange attractors
@article{Eckmann1985ErgodicTO, title={Ergodic theory of chaos and strange attractors}, author={Jean-Pierre Eckmann and David Ruelle}, journal={Reviews of Modern Physics}, year={1985}, volume={57}, pages={617-656} }
Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review…
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References
SHOWING 1-10 OF 177 REFERENCES
Strange Attractors, Chaotic Behavior, and Information Flow
- Physics
- 1981
Simple system equations displaying turbulent behavior are reviewed in the light of information theory. It is argued that a physical implementation of such equations is capable of acting as an…
Estimation of the Kolmogorov entropy from a chaotic signal
- Physics
- 1983
While there has been recently a dramatic growth in new mathematical concepts related to chaotic systems, ' the detailed comparison between models and experimental data has lagged somewhat. After…
The Physics of Chaos and Related Problems: Proceedings of the 59th Nobel Symposium
- Education
- 1985
This symposium was sponsored by the Nobel Foundation through its Nobel Symposium Fund with grants from The Tercentenary Fund of the Bank of Sweden and The Knut $ Alice Wallenberg Foundation.
MEASURES DESCRIBING A TURBULENT FLOW
- Mathematics
- 1980
INTRODUCTION The motion of a fluid in a region R of R' or R' is defined by a function t : , ( t ) , where u ( t ) belongs to some functional space, %, of velocity fields in R. In a turbulent regime,…
Topology of Chaos in a Chemical Reaction
- Physics
- 1981
The authors and coworkers (TURNER et al. [1]; ROUX et al. [2]) have investigated the dynamics of the Belousov-Zhabotinskii reaction in a stirred flow reactor. These experiments, which were motivated…
Gibbsian Distribution on the Lorenz Attractor
- Physics
- 1979
A representation of the Lorentz attractor by a 1-dimensional Ising system is constructed. Gibbsian distribution function which describes the irregular motion of this dissipative dynamical system is…
Large volume limit of the distribution of characteristic exponents in turbulence
- Mathematics
- 1982
For spatially extended conservative or dissipative physical systems, it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity. In…
Intermittency in nonlinear dynamics and singularities at complex times
- Physics
- 1981
High-pass filtering of turbulent velocity signals is known to produce intermittent bursts. This is, as shown, a general property of dynamical systems governed by nonlinear equations with band-limited…