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1999 A genus-5 multispiral attractor

Christophe LETELLIER
06/12/2014
Aziz Alaoui
JPG - 25.9 ko
Aziz Alaoui

Double-scroll attractors were known sinceone was proposed by Otto Rössler  [1] or the Chua circuit was designed but it is possible to produce a larger number of scrolls. This was completed for the first time, at least for the non-autonomous case, by Aziz Alaoui in 1999 [2]. Among other systems he proposed, we here present the one as follows

  \left\{
    \begin{array}{ll}
      \dot{x}&= \alpha \left[y-x-f_3(x)\right] \\
      \dot{y}&= x - y + z \\
      \dot{z}&= -\beta y - \gamma z
    \end{array}
  \right.

where


  f_3(x) = \left|
    \begin{array}{lcl}
      m_0x+\mbox{sgn}(x)\xi_0 & & |x|\leq s_0 \\[0.1cm]
      m_1x+\mbox{sgn}(x)(m_0-m_1)s_0 & \mbox{ if } & 
      s_0\leq |x|\leq s_1 \\[0.1cm]
      m_2x+\mbox{sgn}(x)\left[(m_1-m_2)s_1 +(m_0-m_1)s_0\right] & & s_1 \leq |x|
    \end{array}
  \right.

which produces a genus-5 multispiral attractor as shown in Fig. 1 for parameter values

  \left\{
    \begin{array}{lll}
      \alpha=14,6 &
       \beta=12 &
        \gamma=0.9 \\
   s_0=1 &
   s_1=3 \\
   m_0=-5/7 & m_1=-8/7 & m_2=-0.7
    \end{array}
  \right.

The attractor is here shown in the x-\dot{x} plane projection. Its topological analysis is provided in [3].

JPG - 74.1 ko
Fig. 1. The genus-5 multiscroll attractor.

[1] O. E. Rössler, Continuous chaos,. In : Haken H, editor. Synergetics, Proceedings of the international workshop on synergetics at schloss Elmau (Bavaria, 2–7 May 1977). Springer-Verlag, pp. 184-197, 1977.

[2] M. A. Aziz-Alaoui, Differential equations with multispiral attractors, International Journal of Bifurcation & Chaos, 9, 1009-1039, 1999.

[3] M. Rosalie & C. Letellier, Toward a general procedure for extracting templates from chaotic attractors bounded by high genus torus, International Journal of Bifurcation & Chaos, 24 (4), 1450045, 2014.

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