TURNING-POINT PROPERTIES AS A METHOD FOR THE CHARACTERIZATION OF THE ERGODIC DYNAMICS OF ONE-DIMENSIONAL ITERATIVE MAPS

Dynamical as well as statistical properties of the ergodic and fully d eveloped chaotic dynamics of iterative maps are investigated by means of a turning point analysis. The turning points of a trajectory are he reby defined as the local maxima and minima of the trajectory. An exam ination of the turning point density directly provides us with the inf ormation of the position of the fixed point for the corresponding dyna mical system. Dividing the ergodic dynamics into phases consisting of turning points and nonturning points, respectively, elucidates the und erstanding of the organization of the chaotic dynamics for maps. The t urning point map contains information on any iteration of the dynamica l law and is shown to possess an asymptotic scaling behaviour which is responsible for the assignment of dynamical structures to the environ ment of the two fixed points of the map. Universal statistical turning point properties are derived for doubly symmetric maps. Possible appl ications of the observed turning point properties for the analysis of time series are discussed in some detail. (C) 1997 American Institute of Physics.