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<H1><A NAME="SECTION00020000000000000000">Introduction</A></H1>
<P>
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 [<A HREF="citation.html#tisean" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#tisean">1</A>]. 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 [<A HREF="citation.html#KantzSchreiber" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#KantzSchreiber">2</A>] and the
recent review [<A HREF="citation.html#habil" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#habil">3</A>], 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.
<P>
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 [<A HREF="citation.html#KaplanGlass" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#KaplanGlass">4</A>] 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 [<A HREF="citation.html#Ott" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#Ott">5</A>], but also in
the older books by Bergé et al. [<A HREF="citation.html#Berge" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#Berge">6</A>] and by
Schuster [<A HREF="citation.html#Schuster" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#Schuster">7</A>]. More advanced material is contained in the work by
Katok and Hasselblatt [<A HREF="citation.html#KatokHasselblatt" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#KatokHasselblatt">8</A>]. A collection of research
articles compiled by Ott et al. [<A HREF="citation.html#coping" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#coping">9</A>] covers some of the more applied
aspects of chaos, like synchronization, control, and time series analysis.
<P>
Nonlinear time series analysis based on this theoretical paradigm is described
in two recent monographs, one by Abarbanel [<A HREF="citation.html#abarbook" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#abarbook">10</A>] and one by Kantz and
Schreiber [<A HREF="citation.html#KantzSchreiber" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#KantzSchreiber">2</A>]. While the former volume usually <EM>assumes</EM>
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. [<A HREF="citation.html#gss" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#gss">11</A>], Abarbanel et al. [<A HREF="citation.html#abarbanel" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#abarbanel">12</A>], as well as Kugiumtzis et
al. [<A HREF="citation.html#kugiumtzis_rev1" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#kugiumtzis_rev1">13</A>, <A HREF="citation.html#kugiumtzis_rev2" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#kugiumtzis_rev2">14</A>]. 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 [<A HREF="citation.html#habil" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#habil">3</A>]. Apart from
these works, a number of conference proceedings volumes are devoted to chaotic
time series, including Refs. [<A HREF="citation.html#Mayer-Kress" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#Mayer-Kress">15</A>, <A HREF="citation.html#casdagli" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#casdagli">16</A>, <A HREF="citation.html#SFI" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#SFI">17</A>, <A HREF="citation.html#dyndis" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#dyndis">18</A>, <A HREF="citation.html#freital" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#freital">19</A>].
<P>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan 6 15:38:27 CET 1999</I>
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