~ ) Pergamon Chaos, Solitons & Fractals, Vol.4, No. 1, pp. 133-173,1994 ElsevierScience Ltd Printed in GreatBritain. All rightsreserved 0960-0779/9456.00 + .00 A Comparative Classification of Complexity Measures R. WACKERBAUER 1, A. WITT 2, H. ATMANSPACHER 1, J. KURTHS 2, H. SCHEINGRABER 1 t Max-Planck-Institut fiir extraterrestrische Physik, Giessenbachstrage, DW-8046 Garching 2 Arbeitsgruppe Nichtlineare Dynamik der Max-Planck-Gesellschaft an der Universit~t Potsdam, Am Neuen Palais, DO-1571 Potsdam Abstract - A number of different measures of complexity have been described, discussed, and applied to the logistic map. A classification of these measures has been proposed, dis- tinguishing homogeneous and generating partitions in phase space as well as structural and dynamical elements of the considered measure. The specific capabilities of particular measures to detect particular types of behavior of dynamical systems have been investi- gated and compared with each other. 1. INTRODUCTION 1.1 Complexity The notion of complexity has been object of numerous and extensive studies since it has become clear that the exact sciences, in particular physics, can no longer afford to disregard the behavior of systems which cannot be treated simply. A simple treatment has always been assumed to be possible either if only few degrees of freedom are involved or if central limit theorems can be applied in case of many degrees of freedom. These assumptions cannot be maintained for nonlinear dynamical systems in general. In such systems, complex (in contrast to simple) behavior can occur with only few degrees of freedom, and central limit theorems are not always applicable. A very clear and suggestive illustration of a basic issue arising in the context of defining complexity is due to Grassberger [1]. It is reproduced in Figure 1 and it shows three patterns corresponding to a 133