Statistical analysis and spectral estimation techniques for one-dimensional chaotic signals

Signals arising out of nonlinear dynamics are compelling models for a wide range of both natural and man-made phenomena. In contrast to signals arising out of linear dynamics, extremely rich behavior is obtained even when we restrict our attention to one-dimensional (1-D) chaotic systems with certain smoothness constraints. An important class of such systems are the so-called Markov maps. We develop several properties of signals obtained from Markov maps and present analytical techniques for computing a broad class of their statistics in closed form. These statistics include, for example, correlations of arbitrary order and all moments of such signals. Among several results, we demonstrate that all Markov maps produce signals with rational spectra, and we can therefore view the associated signals as "chaotic ARMA processes," with "chaotic white noise" as a special case. Finally, we also demonstrate how Markov maps can be used to approximate to arbitrary accuracy the statistics any of a broad class of non-Markov chaotic maps.