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

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.