Conditional Entropy of Nonlinear Dynamical Systems

. We present a method for detecting and quantifying relationships between two ore more time series. It is based on the estimation of conditional entropies. In the case of continuous systems an appropriate conditional probability is calculated. Our method can be used for deciding, if one observable Y depends on another one X . If Y can be expressed as a continuous function f(X), the conditional entropy vanishes. These facts help to find optimal delay times in the reconstruction of nonlinear dynamics. Furthermore, we show the advantage of variable delay times. INTRODUCTION For the analysis of nonlinear dynamical systems interdependences between different time sequences X and Y are of special interest. Covariance and mutual information are frequently used for quantifying such relationships [1], [2], [3]. But they are symmetric and therefor one cannot decide if an observable is completely determined by another one. The conditional entropy H(Y jX) seems to be an appropriate measure [4]. I..

Chaotic sonoluminescence

Sonoluminescence (SL) is generated from single, stably oscillating bubbles in a stationary, time-periodic acoustic field. By measuring the time delay between flashes, the dynamics of the phenomenon has been investigated. While other researchers have concentrated on the remarkable periodic stability of the system, present results indicate that, for small variations in the governing parameters, period doubling, chaos, and quasiperiodicity can occur. The implications of the complex temporal behavior for the problem of determining the mechanism(s) for production of SL are discussed.Work was done with partial support from NASA, Office of Naval Research and the Naval Postgraduate School research program, and Deutsche ForschungsgemeinscchaftApproved for public release; distribution is unlimited