Phase relationships between two or more interacting processes from one-dimensional time series. I. Basic theory

A general approach is developed for the detection of phase relationships between two or more different oscillatory processes interacting within a single system, using one-dimensional time series only. It is based on the introduction of angles and radii of return times maps, and on studying the dynamics of the angles. An explicit unique relationship is derived between angles and the conventional phase difference introduced earlier for bivariate data. It is valid under conditions of weak forcing. This correspondence is confirmed numerically for a nonstationary process in a forced Van der Pol system. A model describing the angles' behavior for a dynamical system under weak quasiperiodic forcing with an arbitrary number of independent frequencies is derived.