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Lyapunov Exponents pp 338–360Cite as

Lyapunov exponents and phase transitions in dynamical systems

  • Part V: Chaos And Phase Transitions
  • Conference paper
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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1186))

Abstract

We develop an analogy between the transitions to chaos observed in dissipative dynamical systems and critical phenomena. We mainly focus on the most popular scenarios to weak turbulence, namely the cascade of period-doubling bifurcations and intermittency. We consider the (largest) Lyapunov characteristic exponent as a "(dis)order" parameter which displays scaling behavior in the vicinity of the onset of chaos. We use renormalization group techniques to calculate the corresponding critical exponents. In the presence of an external perturbation, one can carry on the analogy with second-order phase transitions and define cross-over exponents relating the effect of the external field on the Lyapunov characteristic exponent. We compare the effects of both a random noise and a periodic excitation. We report numerical results on discrete systems which corroborate the renormalization group predictions.

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Author notes

  1. A. Arneodo

    Present address: Laboratoire de Physique Théorique, Université de Nice, Parc Valrose, 06034, Nice - Cedex, France

Authors and Affiliations

  1. Centre de Recherche Paul Pascal, Université de Bordeaux I Domaine Universitaire, 33405, Talence - Cedex, France

    F. Argoul & A. Arneodo

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  1. F. Argoul
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Ludwig Arnold Volker Wihstutz

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© 1986 Springer-Verlag

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Argoul, F., Arneodo, A. (1986). Lyapunov exponents and phase transitions in dynamical systems. In: Arnold, L., Wihstutz, V. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1186. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076852

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  • DOI: https://doi.org/10.1007/BFb0076852

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  • Print ISBN: 978-3-540-16458-6

  • Online ISBN: 978-3-540-39795-3

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