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1999 The "Chen-Ueta" system

Christophe LETELLIER
10/05/2009
Guanrong Chen & Tetsushi Ueta

In looking for a system that has an algebraic form similar to the one of the Lorenz system but that is not related to it by a diffeormorphism, Guanrong Chen and Tetsushi Ueta discovered the set of three ordinary differential equations [1]


  \left\{
    \begin{array}{l}
      \dot{x} = \sigma (y-x) \\[0.1cm]
      \dot{y}=(R-\sigma) x +Ry-xz \\[0.1cm]
      \dot{z}=-bz+xy
    \end{array}
  \right.

For appropriate parameter values, this system produces a chaotic attractor (Fig. 1) that is topologically equivalent to the Lorenz attractor. These parameter values are R=22.05, \sigma=35 and b=5.

PNG - 52.9 ko
Fig. 1 : Chaotic "Lorenz" attractor solution to the Chen-Ueta system.

For another set of parameter values, that is, for R=25.264, \sigma=35 and b=1, the Chen-Ueta system produces a chaotic attractor (Fig. 2) that is topologically equivalent to the "Burke and Shaw" system. This system does not differ from the other Lorenz-like systems listed in this compilation. It was recently shown that there exists a homothetic transformation between the Chen-Ueta and the Lorenz systems [2].

PNG - 49.1 ko
Fig. 2 : Chaotic "Burke and Shaw" attractor solution to the Chen-Ueta system.

[1] G. Chen & T. Ueta, Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9, 1465-1466, 1999.

[2] A. Algaba, F. Fernández-Sánchez, M. Merino & A. J. Rodríguez-Luis, Chen’s attractor exists if Lorenz repulsor exists : The Chen system is a special case of the Lorenz system, Chaos, 23 (3), 033108, 2013. On line.

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