  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.
