Predicting the Dimension of Strange Attractors

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
(Received 4 January 1994; revised manuscript received 18 April 1994; accepted for publication 1 July 1994)

ABSTRACT

The correlation dimension was calculated for a collection of 6080 strange attractors obtained numerically from low-degree polynomial, low-dimensional maps and flows. It was found that the average correlation dimension scales approximately as the square root of the dimension of the system with a surprisingly small variation. This result provides an estimate of the number of dynamical variables required to characterize an experiment in which a strange attractor has been found as well as an estimate of the dimension of attractors produced by chaotic systems in which the dimension of the state space is known.

Ref: J. C. Sprott, Physics Letters A 192, 355-360 (1994)

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Fig. 1. The average correlation dimension of 6080 chaotic attractors scales approximately as the square root of the dimension of the system for low-order polynomial maps (O) and flows (X). The error bars represent the spread in dimensions, not an uncertainty in the average values.
[Figure 1]

Fig. 2. The correlation dimension has a high probability of a value about 0.81 times the square root of the dimension of the system.
[Figure 2]

Fig. 3. The average Lyapunov exponent of 6080 chaotic attractors scales approximately inversely with the dimension of the system for low-order polynomial maps (O) and flows (X). The error bars represent the spread in exponents, not an uncertainty in the average values.
[Figure 3]

Fig. 4. The average value of the largest Lyapunov exponent for chaotic maps has a relatively uniform probability when multiplied by the system dimension.
[Figure 4]