Detecting Chaotic Drive–response Geometry in Generalized Synchronization

Dynamical systems which can be decomposed into a drive–response pair sometimes display the phenomenon of generalized synchronization wherein the response state y is a function of the drive state x, y = ϕ (x). The questions of detecting when this is true in experimental data, and what are the properties of the function ϕ, require the development of statistics that are based on concepts and definitions from analysis, rather than the standard statistics literature. We show that these statistics can display when such a generalized synchronization situation exists and characterize the nature of the manifold (given by ϕ) of the response over the drive, including when that manifold is fractal.