The Rossler System

A model dynamical system which is simpler than the Lorenz model was proposed by Rossler. The system consists of three coupled first-order differential equations

$$\dot{x} = -(z + y)$$ (3.14)

$$\dot{y} = x + a y$$ (3.15)

$$\dot{z} = b + x z - c z$$ (3.16)

The system has three degrees of freedom represented by the dynamical variables $x$, $y$ and $z$. Notice the nonlinear term $xz$ in the last equation. There are also three parameters $a, b$ and $c$. By fixing two of the parameters and varying the third one can study the approach to chaos. Let us fix $a=b=0.2$ and treat $c$ as the control parameter. Instead of plotting the time series for the system, it is useful to consider phase portraits, that is by plotting one degree of freedom against another. Figure(3.11) shows the $x-y$ phase portraits (actually only the post-transient orbits are shown) for various values of $c$. One sees clearly period one, period two and period four trajectories: The period here refers to the number of times the cycle goes around before closing. These period doubling sequences lead eventually a chaotic state shown in Figure(3.11d).

The behaviour of the $z$ variable in the chaotic state ($c=5$) is illuminated by looking at the time series shown in Figure(3.12a). From that one sees that the $z$ coordinate is small most of the time, meaning that the trajectories mostly are close to the $x-y$ plane. However occasionally there are large spikes in the $z$ time series leading to a jump in the $z$ coordinate. For reference the time series for the $x$ variable is shown in Fig(3.12b).

Fig.(3.12a)

Fig.(3.12b)

The full three dimensional phase portrait for the chaotic state shown in Figure (3.13). This is the strange attractor for the system. Notice that the attractor lies in a bounded region of phase space, and its structure, though complicated, is far from random. Indeed strange attractors have been shown to have a fractal structure, that is self-similarity at different scales! This is our first indication that deterministic chaos is different from noise or randomness.

There is an apparent paradox about chaotic states. We noted that they were characterized by a sensitive dependence on initial conditions, which leads nearby trajectories to diverge exponentially . On the other hand the phase portrait is bounded: This is why the strange attractor justifies its ‘attractor’ label. So how are trajectories repelled and attracted at the same time ? This is done by a process of stretching of nearby trajectories on short time scales, followed by a process of folding at longer scales. The folding process is evident in Figs(3.11d) and (3.13): Follow some trajectory in the $z=0$ plane, it will soon be lifted up at a spike of the $z$-component, and is then folded back into the $z=0$ plane of the attractor at another point, to continue its evolution. The stretching and folding processes mix nearby trajectories in the attractor and results in a decorrelation of closeby initial states.