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1981 The Burke & Shaw system

Christophe LETELLIER
29/01/2008
Bill Burke & Robert Shaw


- 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.

where S and V are the parameters. This system is invariant under a rotation symmetry {\cal R}_z 
(\pi) around the z-axis. For (S,V)=(10,4.272), a chaotic attractor is obtained (Fig. 1).

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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 z axis [2]. This attractor is characterized by a four branch template (Fig. 2) [3].

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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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