1999 A genus-5 multispiral attractor
- 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.](local/cache-TeX/84bc53d82a03d68e0e73bcdc7c61d2a5.png "\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.](local/cache-TeX/3a382b93ac5331867f8b2caa1ec8c892.png "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
The attractor is here shown in the 
-
plane projection.
Its topological analysis is provided in [3].
- 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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MàJ . 04/02/2024