1980 The Shimizu-Morioka system

A system algebraically simpler than the Lorenz system has been proposed by Shimizu & Morioka [1].

 The system

The set of three ordinary differential equations known as the "Shimizu & Morioka" system reads as :

This system has one fixed point, \(F_0\), located at the origin of the phase space and two fixed points\(F_\pm\) located at \((\pm \sqrt{\alpha},0,1)\). For a wide range of
parameter values, including those corresponding to a chaotic attractor,
\(F_0\) is a saddle and \(F_\pm\) are two saddle-foci.
This system produces a Lorenz-like'' chaotic attractor with parameter values <math>$ \mu=0.81$</math> and <math>$ \alpha=0.375$</math> (Fig. 1). <doc165|center> Replacing the <math>$\alpha$</math> with 0.191450 changes the attractor for aBurke and Shaw-like’’ attractor (Fig. 2).