Ergodic theory of chaos and strange attractors

Abstract

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 is an account of the main mathematical ideas and, their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part I: Stationary Case

“…The stochastic generator used in this paper to produce the time series is a deterministic model (an ordinary differential equation), whose dynamics, for the considered range of values of T E , is chaotic in the sense that it takes place on a strange attractor Λ in phase space (Eckmann and Ruelle 1985). See Lucarini et al (2006c,d) for a study of the properties of this attractor, including sensitivity with respect to initial conditions.…”

“…See Lucarini et al (2006c,d) for a study of the properties of this attractor, including sensitivity with respect to initial conditions. The statistical behaviour of this type of time series is determined by the Sinai-Ruelle-Bowen (SRB) probability measure µ (Eckmann and Ruelle 1985): this is a Borel probability measure in phase space which is invariant under the flow f t of the differential equation, is ergodic, is singular with respect to the Lebesgue measure in phase space and its conditional measures along unstable manifolds are absolutely continuous, see Young (2002) and references therein. Moreover, the SRB measure is also a physical measure: there is a set V having full Lebesgue measure in a neighbourhood U of Λ such that for every continuous observable φ : U → R, we have, for every x ∈ V , the frequency-limit characterization…”

Thresholds in transient dynamics of signal transduction pathways

“…Lyapunov exponents measure the exponential divergence between a reference trajectory and d orthogonal perturbations to the trajectory, where d is the dimension of the related mathematical model which is equal to the number of state variables [21]. Finite time Lyapunov exponents (FTL) have been introduced to quantify dynamical instabilities over a finite interval of time [22,23,24,25].…”

Methods for short time series analysis of cell-based biosensor data

“…We accomplished this by taking the measured time series, s(t), and creating d E dimensional vectors by using the well-known time delay method (Takens, 1981;Eckmann and Ruelle, 1985;Eckmann et al, 1986;Sauer et al, 1991). Theory states that by doing this construction for a deterministic process, the resulting picture is equivalent (topologically) to measuring all independent variables, such as state space variables and their time derivatives.…”