Strange attractor definition

An example of a calculation of the Lyapunov exponents and dimension, for a simple four-variable model of the peroxidase-oxidase reaction will help to clarify these general definitions. The following material is adapted from the presentation in Ref. 94. As described earlier, the Lyapunov dimension and the correlation dimension, D, serve as upper and lower bounds, respectively, to the fractal dimension of the strange attractor. The simple four-variable model is similar to the Degn—Olsen-Ferram (DOP) model discussed in a previous section but was suggested by L. F. Olsen a few years after the DOP model was introduced. It remains the simplest model the peroxidase-oxidase reaction which is consistent with the most experimental observations about this reaction. The rate equations for this model are ... 

The Liapunov number can be used as a quantitative measure for chaos. The connection between chaos and the Liapunov number is through attractors. An attractor is a set of points S such that for nearly any point surrounding S, the dynamics will approach S as the time approaches infinity. The steady state of a fluid flow can be termed an attractor with dimension zero and a stable limit cycle dimension one. There are attractors that do not have integer dimensirms and are often called strange attractors. There is no tmiversally acceptable definition for strange attractors. The Liapunov number is determined by the principle axes of the ellipsoidal in the phase space, which originates from a ball of points in the phase space. The relatitaiship between the Liapunov number and the characterization of chaos is not universal and is an area of intensive research. 

In quite a few cases, chaos can be predicted based on appropriate mathematical model (Lorenz or Rosselor) (Chapter 12). The strange attractor obtained in such cases signifies Deterministic Chaos . Such models exhibit cross-catalysis and inhibition. Experimentally observed deterministic chaos can be easily ascertained using definite criteria, although prediction of complex time order is quite difficult in many cases. In terms of causality principle, the complexity in the system arises due to complex interdependence of a number of causes and effects. 

Another important property of an attractor is its dimension. It is, loosely speaking, the number of independent degrees of freedom relevant to the dynamical behavior. There are several different definitions that differ mainly in the measure used [35, 39] these definitions all yield the Euclidean value of dimension for Euclidean objects, but for strange attractors the dimension is in general fractional ("fractal [40]). As an example of computing the dimension from experimental data, we will describe a procedure for computing the information dimension, d [41]. Let N(e) be the number of points in a ball of radius e about a point x on an attractor. For a uniform density of points one would have... 

A strange (or chaotic) attractor is by definition an attractor for which the largest Lyapunov exponent is positive. Then trajectories starting from nearby points will separate exponentially fast as time evolves. Therefore, all information about the initial conditions is rapidly lost, since any uncertainty, no matter how small, will be magnified until it becomes as large as the attractor thus there is sensitive dependence on initial conditions (RUELLE [38]). Long term predictions about the state of the system are impossible. 

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