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Stability of the Scenarios Towards Chaos

  • P. Coullet
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 24)

Abstract

For the last ten years, the study of chaotic behaviors generated by deterministic dynamical systems has received an increasing interest among Physicists and Mathematicians. A fascinating aspect of this field of research is its interdisciplinarity. The deep reason for this universality lies in the very nature of the subject: The Mathematics of Time [1]. The study of simple dynamical systems, such as the iterations of mappings or the ordinary differential equations thus appears as a study of these Mathematics of time. Two kinds of problems can be easily identified. The first one deals with the transition towards complex behaviors. The second one is concerned by the chaotic behaviors themselves. In this paper, we will only discuss some general aspects of the first class of problems. In fact, these problems are in some sense simpler to study. They generally lead to universal scenarios [2] by which a periodic or quasiperiodic signal progressively looses its regularity, as an external constraint varies. These problems of transition bear a striking analogy with critical phenomena. The common analytical tool is the renormalization group [3].

Keywords

Chaotic Behavior Critical Phenomenon Stable Manifold Period Doubling Invariant Torus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • P. Coullet
    • 1
  1. 1.Mécanique StatistiqueLaboratoire de Physique de la Matière CondenséeNice CedexFrance

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