Abstract
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.
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References
What is Chaos
For mathematically rigorous treatments of examples of chaos by means of mappings and other topological methods see S. Smale: Bull. A. M. S. 73, 747 (1967)
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Historically, the first papers showing a “strange attractor”. For further treatments of this model see C. T. Sparrow: The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (Springer, Berlin-Heidelberg-New York 1982)
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How Chaos Occurs
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Correlation Function and Frequency Distribution
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This paper deals with the logistic map given in the text. The universal behavior of period doubling was discovered by M. J. Feigenbaum: J. Stat. Phys. 19, 25 (1978):
This paper deals with the logistic map given in the text. The universal behavior of period doubling was discovered by M. J. Feigenbaum: Phys. Lett. A74, 375 (1979)
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H. Haken (ed.): Chaos and Order in Nature, Springer Ser. Synergetics, Vol. 11 (Springer, Berlin Heidelberg, New York 1981)
H. Haken (ed.): Evolution of Order and Chaos, Springer Ser. Synergetics, Vol. 17 (Springer, Berlin, Heidelberg, New York 1982)
The influence of fluctuations on period doubling has been studied by the following authors: G. Mayer-Kress, H. Haken: J. Stat. Phys. 24, 345 (1981)
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The influence of fluctuations on period doubling has been studied by the following authors: B. Shraiman, C. E. Wayne, P. C. Martin: Phys. Rev. Lett. 64, 935 (1981)
The corresponding Kolmogorov equation is established and discussed in H. Haken, G. Mayer-Kress: Z. Phys. B43. 185 (1981)
The corresponding Kolmogorov equation is established and discussed in H. Haken, A. Wunderlin: Z. Phys. B46, 181 (1982)
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Haken, H. (1983). Chaos. In: Synergetics. Springer Series in Synergetics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88338-5_12
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