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1981 A possible model for the onset of turbulence

Christophe LETELLIER
03/05/2009
Alain Arnéodo, Pierre Coullet & Charles Tresser

In their study for one route to chaos via a cascade of bifurcations involving homoclinic orbits, Alain Arnéodo, Pierre Coullet et Charles Tresser obtained a set of three ordinary differential equations with a rotation symmetry [1]. The system


  \left\{
    \begin{array}{l}
       \dot{x}=\alpha (x-y) \\[0.1cm]
       \dot{y}=\mu x^3 -4 \alpha y+xz \\[0.1cm]
       \dot{z}=-\delta \alpha z+ xy+ \beta z^2
    \end{array}
  \right.

produces an attractor topologically equivalent to the Lorenz attractor for  \alpha=1.8, \beta=-0.07,  \delta=1.5, \mu=0.02.

PNG - 68.5 ko
Fig. 1 : "Lorenz attractor" solution to the ACT model

Changing the parameters to  \alpha=1.8, \beta=-0.02,  \delta=1.425, \mu=0.0001 leads to an attractor topologically equivalent to a "Burke and Shaw attractor" (Fig. 2).

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Fig. 2 : "Burke and Shaw attractor" solution to the ACT model.

[1] A. Arnéodo, P. Coullet & C. Tresser, A new possible mechanism for the onset of turbulence, Physics Letters A, 81 (4), 197-201, 1981.

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