Phase Transitions Induced by Deterministic Delayed Forces

  • M. Le Berre
  • E. Ressayre
  • A. Talllet
Part of the NATO ASI Series book series (NSSB, volume 208)


The frontier between noise and deterministic chaos was recently shown1 to be free, in the sense that the very simple deterministic retarded equation
$$\frac{{dx}}{{dt}} + \frac{{\partial v}}{{\partial x}}\left( {x\left( t \right)} \right) = \sin \left[ {Ax\left( {t - d} \right)} \right]$$
with ∂V/∂x = x was shown to behave as a linear Langevin equation with Gaussian noise, in the limit A → ∞. In deterministic systems the statistics of the feedback term cannot be stated a priori, it follows from internal properties of the equation. Let us point out that the Gaussian behavior of x results from the analytical form choosen for the feedback f(x) in the sense that the periodic character of f(x) is responsible for the short memory effects in f[ax(t)]. On the contrary, in the case of Mackey-Glass equation2, the feedback will never get short memory as the parameter A increases, because the corresponding feedback f(x) = x / 1+xc has only one maximum.


Deterministic Chaos Short Memory Feedback Term Gaussian Behavior Noise Induce Transition 
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  1. 1.
    a) B. Dorizzi, B. Grammaticos, M. Le Berre, Y. Pommeau, E. Ressayre, A. Tallet, Phys. Rev. A 35, 328, 1987 ;Google Scholar
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    M. Le Berre, E. Ressayre, A. Tallet, Proc. of IQEC Conference in Rochester, June 1989.Google Scholar
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    See for example J. Masoliver, B.J. West, H. Lindenberg, Phys. Rev. A 35, 3086, 1987 and references herein.Google Scholar
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    W. Horsthemke, R. Lefever “Noise induced transitions : Theory and applications in Physics, Chemistry and Biology”, Springer Verlag, 1984. See also R.R. Grigolini, L.A. Lugiato, R. Mannella and P.V.E. McClintock, M. Meni, M. Pernigo, Phys. Rev. A38, 1966, 1988 and references herein.Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • M. Le Berre
    • 1
  • E. Ressayre
    • 1
  • A. Talllet
    • 1
  1. 1.Laboratoire de Photophysique MoléculaireUniversité Paris-SudOrsayFrance

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