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Mathematical Background

  • Chapter
Encounter with Chaos

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Abstract

The interest in nonlinear dynamical systems increased considerably when it was realized that very modest model cases could explain a whole range of dynamical behavior, from simple to very complex. In his pioneering work of 1963, E. Lorenz showed that his simple system of nonlinear differential equations describing a forced dissipative hydrodynamical flow was able to produce, in addition to regular behavior, aperiodic bounded solutions and discontinuous dependence on initial conditions. Earlier it had been believed that a description of what was called “turbulent motion” necessarily involved a large number of degrees of freedom, if a different interpretation from noise could be given at all. Now, suddenly, it was observed that a whole class (often called a “universality class”) of apparently different theoretical and experimental systems behaved alike, when an “external” parameter changed in such a way that the system was led from simple to more complex behavior. The reason for this could be traced back, for both dissipative and nondissipative dynamical systems, from the differential equation to the Poincaré section and from there to a model map. This model map was shown to be characteristic for each universality class; it contains the whole information of how the change from simple to more complex behavior is achieved. In this way it became possible to describe two “generic” ways for this metamorphosis: period doubling and the quasiperiodic route to chaos. A representative collection of early works is furnished in [4.1].

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Authors and Affiliations

  1. Centre de Recherche sur les très Basses Températures, C.N.R.S., PB 166, F-38042, Grenoble Cedex 9, France

    Dr. Joachim Peinke

  2. Physics Institute, University of Zurich, Schönberggasse 9, CH-8001, Zurich, Switzerland

    Priv.-Doz. Dr. Jürgen Parisi & Dr. Ruedi Stoop

  3. Institute for Physical and Theoretical Chemistry, University of Tübingen, Morgenstelle 8, W-7400, Tübingen, Fed. Rep. of Germany

    Prof. Dr. Otto E. Rössler

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  1. Dr. Joachim Peinke
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  2. Priv.-Doz. Dr. Jürgen Parisi
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  3. Prof. Dr. Otto E. Rössler
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Peinke, J., Parisi, J., Rössler, O.E., Stoop, R. (1992). Mathematical Background. In: Encounter with Chaos. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77625-0_4

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