ESTIMATION OF THE CORRELATION DIMENSION OF DYNAMICAL SYSTEM ATTRACTORS BASED ON MATRIX COMPUTATIONS (CDM - METHOD)
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Abstract
The methods for estimating the correlation dimension of attractors of dynamical systems, an invariant measure of the heterogeneity of the fractal structure of attractors, have been studied.
The aim of the article is to develop effective methods for estimating the correlation dimension of attractors in dynamical systems and to apply them to a range of models, enabling a comparison of the obtained results with theoretical and experimental findings.
Based on the Grassberger–Procaccia algorithm, a method for calculating the correlation dimension using matrix computations has been developed and tested. This method involves constructing a distance matrix between points of the attractor in phase space, sorting the matrix, and analyzing the logarithmic dependency graph. The developed methods have been applied to «reference» attractors: the logistic map, Henon map, Kaplan–Yorke attractor, Zaslavsky attractor, Lorenz, Rössler and Rabinovich–Fabrikant systems. This application confirmed theoretical aspects through experimental results. The use of matrix-based computations has enabled not only an increase in the calculation speed but also improved the accuracy of the estimates, as the method accounts for all points of the attractor.
The study results indicate that the three-dimensional attractors of Lorenz, Rössler, and Rabinovich–Fabrikant systems have a correlation dimension within the range , aligning with theoretical predictions and consistent with previous estimates. Meanwhile, the correlation dimension for shuffled data is significantly higher and approaches the dimensionality of the phase space, indicating the destruction of the attractor's delicate fractal structure.
The described methods generalize the research conducted by the authors and can be applied to a wide range of problems in physical and economic dynamics. The article contributes to the development of methods for assessing the fractal characteristics of attractors and expanding their application in scientific studies.
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