Part of the Springer Series in Synergetics book series (SSSYN, volume 1)
Sometimes scientists like to use dramatic words of ordinary language in their science and to attribute to them a technical meaning. We already saw an example in Thorn’s theory of “catastrophes”. In this chapter we become acquainted with the term “chaos”. The word in its technical sense refers to irregular motion. In previous chapters we encountered numerous examples for regular motions, for instance an entirely periodic oscillation, or the regular occurrence of spikes with well-defined time intervals. On the other hand, in the chapters about Brownian motion and random processes we treated examples where an irregular motion occurs due to random, i.e., in principle unpredictable, causes. Surprisingly the irregular motion represented in Fig. 12.1 stems from completely deterministic equations. To characterize this new phenomenon, we define chaos as irregular motion stemming from deterministic equations. The reader should be warned that somewhat different definitions of chaos and the criteria to check its occurrence are available in the literature. The difficulty rests mainly in the problem of how to define “irregular motion“properly. For instance, the superposition of motions with different frequencies could mimic to some extent an irregular behavior and one wants to preclude such a case from representing chaos. We shall come back to this question in Section 12.5, where we shall discuss the typical behavior of the correlation function of chaotic processes. A good deal of present-day analysis of “chaos” rests on computer calculations.
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Chaos 12.1 What Is Chaos?
The Lorenz Model. Motivation and Realization
Chaos and the Failure of the Slaving Principle
- H. Haken, J. Zorell: UnpublishedGoogle Scholar
Correlation Function and Frequency Distribution
- Y. Aizawa, I. Shimada: Preprint 1977Google Scholar
Further Examples of Chaotic Motion
- Three-Body Problem:Google Scholar
- H. Poincaré: Les méthodes nouvelles de la méchanique céleste. Gauthier-Villars, Paris (1892/99), Reprint (Dover Publ., New York 1960)Google Scholar
- Gunn Oscillator:Google Scholar
- O. E. Roessler. For a summary and list of references consultGoogle Scholar
- O. E. Roessler: In Synergetics, A Workshop, ed. by H. Haken (Springer, Berlin-Heidelberg-New York, 1977)Google Scholar
© Springer-Verlag Berlin Heidelberg 1978