Predicting the dimension of strange attractors

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. It has become fashionable to search for simple determinism (chaos) in fluctuating, non-periodic, experimental data. This is often done by calculating the correlation dimension [ 1 ] from a time-series record using the method of time-delay reconstruction [ 2,3 ]. This method has been applied to systems as diverse as the stock market [ 4 ], sunspots [ 5 ], rainfall [ 6 ], electrocardiograms [ 7 ], electroencephalograms [ 8 ], and childhood epidemics [9]. Such studies are motivated by the hope that a strange attractor with low fractional dimension will be found, in which case it might be possible to model the dynamics using a number of variables as small as the next higher integer. It is useful to consider whether the dimension of an attractor provides any additional information about the dimension of the system that produced it. This paper addresses this question by calculating the distribution of correlation dimensions of strange attractors produced by various systems of equations. This novel statistical approach should provide guidance in modeling chaotic data and in searching for low-dimensional attractors in systems whose statespace dimension is known. Consider first the case of general iterated N-th degree polynomial/)-dimensional maps given by