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1980 The Shimizu-Morioka system

Christophe LETELLIER
20/05/2009
T. Shimizu & N. Morioka

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 :

  \left\{
  \begin{array}{l}
      \dot{x}=y \\[0.1cm]
      \dot{y}=x-\mu y-xz \\[0.1cm]
      \dot{z}=-\alpha z+x^2
   \end{array}
  \right.  \, .
This system has one fixed point, F_0, located at the origin of the phase space and two fixed pointsF_\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  \mu=0.81 and  \alpha=0.375 (Fig. 1).

PNG - 70.1 ko
Fig. 1 : Lorenz-like chaotic attractor solution to the Shimizu-Morioka system

Replacing the \alpha with 0.191450 changes the attractor for a ``Burke and Shaw-like’’ attractor (Fig. 2).

PNG - 59.5 ko
Fig. 2 : Burke-Shaw-like attractor solution to the Shimizu-Morioka system.

[1] T. Shimizu & N. Moroika, On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model, Physics Letters A, 76, 201-204, 1980.

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