Deterministic chaos as a fundamental concept is by now well established and described in a rich literature. The mere fact that simple deterministic systems generically exhibit complicated temporal behavior in the presence of nonlinearity has influenced thinking and intuition in many fields. However, it has been questioned whether the relevance of chaos for the understanding of the time evolving world goes beyond that of a purely philosophical paradigm. Accordingly, major research efforts are dedicated to two related questions. The first question is if chaos theory can be used to gain a better understanding and interpretation of observed complex dynamical behavior. The second is if chaos theory can give an advantage in predicting or controlling such time evolution. Time evolution as a system property can be measured by recording time series. Thus, nonlinear time series methods will be the key to the answers of the above questions. This paper is intended to encourage the explorative use of such methods by a section of the scientific community which is not limited to chaos theorists. A range of algorithms has been made available in the form of computer programs by the TISEAN project [1]. Since this is fairly new territory, unguided use of the algorithms bears considerable risk of wrong interpretation and unintelligible or spurious results. In the present paper, the essential ideas behind the algorithms are summarized and pointers to the existing literature are given. To avoid excessive redundancy with the text book [2] and the recent review [3], the derivation of the methods will be kept to a minimum. On the other hand, the choices that have been made in the implementation of the programs are discussed more thoroughly, even if this may seem quite technical at times. We will also point to possible alternatives to the TISEAN implementation.
Let us at this point mention a number of general references on the subject of nonlinear dynamics. At an introductory level, the book by Kaplan and Glass [4] is aimed at an interdisciplinary audience and provides a good intuitive understanding of the fundamentals of dynamics. The theoretical framework is thoroughly described by Ott [5], but also in the older books by Bergé et al. [6] and by Schuster [7]. More advanced material is contained in the work by Katok and Hasselblatt [8]. A collection of research articles compiled by Ott et al. [9] covers some of the more applied aspects of chaos, like synchronization, control, and time series analysis.
Nonlinear time series analysis based on this theoretical paradigm is described in two recent monographs, one by Abarbanel [10] and one by Kantz and Schreiber [2]. While the former volume usually assumes chaoticity, the latter book puts some emphasis on practical applications to time series that are not manifestly found, nor simply assumed to be, deterministic chaotic. This is the rationale we will also adopt in the present paper. A number of older articles can be seen as reviews, including Grassberger et al. [11], Abarbanel et al. [12], as well as Kugiumtzis et al. [13, 14]. The application of nonlinear time series analysis to real world measurements where determinism is unlikely to be present in a stronger sense, is reviewed in Schreiber [3]. Apart from these works, a number of conference proceedings volumes are devoted to chaotic time series, including Refs. [15, 16, 17, 18, 19].