3.5 Comparison with the Lorenz Equation

The familiar Lorenz equation is given by

$$\begin{array}{rcl} \displaystyle \frac{dx}{dt} & = & a (y - ... ...kip0.5ex} \displaystyle \frac{dz}{dt} & = & xy - bz \end{array}$$ (18)

Many research reports [Jackson, 1991; Argyris, 1994] mentioned its bifurcation structure in the $c$-$b$ plane. We now show its bifurcation sets in the $a$-$c$ plane in comparison with Chen's equation.

This system also has three equilibria: $O$, $C^{+}$, and $C^{-}$, and their topological properties are almost the same as that of Chen's equation (7). Figure 16 shows the bifurcation diagram of solutions of the Lorenz system (18). There exist islands of period-locking, which are the same as those shown in Fig. 2. The chaotic attractor, as shown in Fig. 17 (g), is widely observed in this parameter plane.

Figure 16: Bifurcation diagram of the Lorenz system (18). $c=28$.

Figure 17: Phase portrait of Eq. (18). $c=28$. (a) $b=0.6$, (b) $b=0.85$, (c) $b=0.94$, (d) $b=0.98$, (e) $b=1.05$, (f) $b=1.1$, (g) $b = 1.15$.

Hopf bifurcations at the equilibria, except for the origin, occurs at the value of

$$a_h = \frac{-3-b-c + \sqrt{((3+b-c)^2 -4(1+b)c)}}{2}$$ (19)

When $a < a_h$, the system orbit, starting from the neighborhood of $C^{+}$ or $C^{-}$, is trapped into the chaotic attractor. While if $a> a_h$, the chaotic attractor is eventually absorbed by $C^{+}$ or $C^{-}$. It is the well-known ``crisis'' phenomenon.

The stable manifolds of $C^{\pm}$ are very simple, as can be seen from Fig. 18. The mechanism of chaos is therefore completely different from Chen's strange attractor.

Figure 18: Stable manifolds of $C^{+}$ and $C^{-}$. $a=45$, $b=2.8$, $c=28$.