telech

1994 Unimodal toroidal chaos

Christophe LETELLIER
03/05/2009
Bo Deng
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Bo Deng

Toroidal chaos is quite rare in 3D autonomous continuous systems. Apart the toroidal system produced by Otto Rössler in 1979 but which does not produce chaotic behaviors, the system by Bo Deng could be one of the very first to produce toroidal chaos. What is quite interesting in this attractor is that there is a unique folding mechanism : this is thus unimodal chaos.

- The system

This is a set of three ordinary differential equations reading as [1]


\left\{
  \begin{array}{l}
     \dot{x} = z (\lambda x - \mu y )
      + (2-z) \left[ \alpha x  \left( \displaystyle 1-\frac{x^2+y^2}{R^2} \right) -\beta y \right] \\[0.4cm]
      \dot{y} = z ( \mu x +\lambda y)+
     (2-z) \left[ \alpha y \left( \displaystyle  1- \frac{x^2+y^2}{R^2} \right)+\beta x \right] \\[0.4cm]
      \dot{z}= \frac{1}{\epsilon} \left[z ( (2-z) \left( \displaystyle a (z-2)^2+b \right) - dx)
      \left(z+m \left( \displaystyle x^2+y^2 \right)-\eta \right)-\epsilon c(z-1) \right]
    \end{array}
  \right. \, .
This system produces a toroidal chaotic attractor (Fig.1) with parameter values as a=3, b=0.8, c=1, d=0.1, m=0.05,  \eta=3.312, R=10, \alpha=2.8, \beta=5, \epsilon=0.1, \lambda=-2 and \mu=1. A sketch of the attractor is shown in Fig. 2.

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Fig. 1 : Toroidal chaos with a single fold
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Fig. 2 : Skectch of the toroidal chaotic attractor

A Poincaré section revealing the single folding is shown in Fig. 3.

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Fig. 3. Poincaré section of the unimodal toroidal chaotic attractor

[1] Bo Deng, Constructing homoclinic orbits and chaotic attractors, International Journal of Bifurcation & Chaos, 4, 823-841, 1994.

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