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 \(\alpha=1.8\), \(\beta=-0.07\), \(\delta=1.5\), \(\mu=0.02\).
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).
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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