Nonlinear Sciences – Adaptation and Self-Organizing Systems
Scientific paper
2002-12-23
Nonlinear Sciences
Adaptation and Self-Organizing Systems
26 pages, 20 figures, Working paper-comments and inquiries welcome
Scientific paper
An inverse modeling technique is introduced that combines elements of coupled logistic map models and wavelet analysis for the purpose of analyzing partial synchronization states in high-dimensional systems. Using Embedded Complex Logistic Maps (ECLM), time series data derived from individual system components is directly mapped to a wavelet-like space generated from iterations of specific complex logistic maps. These maps are selected from the complex plane according to "best fit" scoring criteria with the data. The embedding topology within and near the familiar Mandelbrot Set provides metrics which are used to aid in clustering similarities (synchronization) between these selected point models. The dynamics within the individual models and the correlation between (synchronized) models is analyzed to reconstruct a unified picture of the underlying dynamics of local system components within the global system network topology. In this paper, ECLM is used to extract system parameters, network graphs, and wavelet-like analytics from two real-world systems, a gene expression network and a financial market network. Preliminary results appear to validate the assumptions and methods used in ECLM through agreement with current theory and recent findings. Some potentially new findings regarding synchronization, scale-free networks, on-off intermittency, and energy dissipation within the examples studied will also be discussed.
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