Nuclear Physics B (Proc. Suppl.) 2 (1987) 527-538 527 North-Holland, Amsterdam EXPLORING DETERMINISTIC CHAOS VIA UNSTABLE PERIODIC ORBITS Itamar Procaccia Department of Chemical Physics, Weizraann Institute of Science, Rehovot 76100, Israel The application of standard dimension algorithms is not always successful in deciding whether a given data set represents deterministic chaotic motion. We propose to supplement such algorithms with a search of the underlying unstable periodic orbits. These appear to give important clues to the organization of strange attractors and to their scaling properties. It is shown how to extract these orbits from experimental data sets, avA how to use them to obtain information on dimensions, entropies avA the spectrum of scaling exponents of the ergodic invariant measure. Finally they are used to obtain approximants for the invariant measure itself. These approximants can be used to calculate a variety of interesting invartants that characterize the chaotic motion. 1. INTRODUCTION The phenom~ological study of chaotic motion has benefitted from the advent of algorithms which were well geared to calculate the dlmev~Ious and entropies of strange attractors. 1 Besides being helpful in controlled studies of physical systems, these algorithms were widely applied to data sets coming from uncontrolled, or partially controlled, system~ from the natural sciences 2 (geology, cosmology etc.), the llfe sciences 3 and economics. 4 In many of these latter applications, ambiguous results concerning the deterministic nature of the tested data sets were obtained. The opinion of this author is that indeed in a number of such cases the data did not originate from deterministic dynamics, and the existence of a finite dimension of an alleged phase-space orbit was an artifact. 5 In this paper I suggest to supplement dimension algorithms with an even more scrutinizing test of deterministic chaos: the existence of underlying unstable periodic orbits. Deterministic chaos is often defined as being tantamount to the existence of a positive Lyapunov exponent. 6 This is known as the condition for "strong chaos. M A weaker condition (which is a necessary condition for strong chaos in generic cases) is the abundance of asymptotic states or a positive topological entropy. In simple words this condition states that the number of periodic orbits of length n increases exponentially with n, the exponent belng7 the topologlcal entropy K . o We shall assume throughout the existence of a discrete time orbit {Xi}~:l._ In cases where continuous time orbits X(t) are given, usual methods of Poincare surfaces of sections are used to obtain discrete time series. The weak condition for chaos is then: 0920-5632/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)