telech

2011 A non hyperchaotic but 4D system

Christophe LETELLIER
08/11/2016

In 2011, Liangrui Tang, Lin Zhao, and Qin Zhang proposed a new four-dimensional system [1]


\left\{
    \begin{array}{l}
      \dot{x} = a (y -x) + yz \\[0.1cm]
      \dot{y} = b (x+y) - xz \\[0.1cm]
      \dot{z} = cx - dz  + yw \\[0.1cm]
      \dot{w} = ey - fw + xz  \, . 
    \end{array}
  \right.

which produces an interesting attractor for the parameter values as a=55, b=25, c=40, d=13, (e=23, and f=8. Initial conditions can be as x = 0, y = 1, z = 0 and w=1. Two planes projections of the attractor are shown in Fig. 1. Contrary to what was initially claimed, this attractor is not hyperchaotic and has a single positive Lyapunov exponent [2].

JPG - 27.7 ko
Fig. 1. Four-dimensional chaotic attractor.

This is confirmed by the one-dimensional first-return map to a Poincaré section which is shown in Fig. 2.

JPG - 12.7 ko
Fig. 2. First-return map to a Poincaré section.

[1] L. Tang, L. Zhao & Q. Zhang, A novel four-dimensional hyperchaotic system, In : \it Applied Informatics and Communication, Springer-Verlag, pp. 392-401, 2011.

[2] J. P. Singh & B. K. Roy, The nature of Lyapunov exponents is (+,+,-,-). Is it a hyperchaotic system ?, Chaos, Solitons & Fractals, 92, 73-85, 2016.

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