1981 The Burke & Shaw system
The system
The Burke & Shaw system has been derived by Bill Burke and Robert Shaw from the Lorenz equations [1] The set of ordinary differential equations is
![\left\{
\begin{array}{l}
\dot{x}=-S(x+y) \\[0.3cm]
\dot{y}=-y-Sxz \\[0.3cm]
\dot{z}=Sxy+V
\end{array}
\right.](local/cache-TeX/268e6a3eaae56cfcc64b2c2167bbf990.png "\left\{
\begin{array}{l}
\dot{x}=-S(x+y) \\[0.3cm]
\dot{y}=-y-Sxz \\[0.3cm]
\dot{z}=Sxy+V
\end{array}
\right.")
where 
and 
are the parameters. This system is invariant under a rotation symmetry 
around the 
-axis. For 
, a chaotic attractor is obtained (Fig. 1).
- Fig. 1: Chaotic attractor
This system is a companion to the Lorenz system, in the sense that it belongs to the same class of systems. The main departure between the Burke & Shaw system and the Lorenz system is not in their equations but in the way they are organized around the axis [2]. This attractor is characterized by a four branch template (Fig. 2) [3].
- Fig. 2: Branched manifold of the Burke & Shaw system
[1] R. Shaw, Strange attractor, chaotic behavior and information flow, Zeitschrift für Naturforsch A, 36, 80-112, 1981.
[2] C. Letellier, T. Tsankov, G. Byrne & R. Gilmore, Large-scale structural reorganization of strange attractors, Physical Review E, 72, 026212, 2005.
[3] C. Letellier, P. Dutertre, J. Reizner & G. Gouesbet, Evolution of multimodal map induced by an equivariant vector field, Journal of Physics A, 29, 5359-5373, 1996.
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MàJ . 04/02/2024