In 1976, Otto Rössler⁽¹⁾ set out to formulate the simplest continuous dynamical system he could thinkcapable of generateing chaotic solutions. The resulting three equations are

dx/dt = -(y + z)

dy/dt = x + a y

dz/dt = b + x z - c z

where a, b and c are parameters. For some parameter values (Figure 1), the asymptotic dynamics lie on what amounts to a Mobius strip with a single twist. The result is what Rössler called "spiral chaos." ⁽²⁾   In greater detail, trajectories on the central disk spiral outward from a saddle focus of index 2.⁽³⁾   Eventually, they rise up in the z-direction and are then folded back down again. The disk itself is composed of an infinite number of layers, but these are so compressed that trajectories based at different initial conditions become effectively identified. The results are effective irreversibility and sensitivity to initial conditions, the hallmark of chaotic motion.

The animation shown here views a single, post-transient solution curve from a succession of angles, giving the illusion of a spinning attractor.

The Rössler attractor "lives" in three dimensions. But because the central disk is essentially two-dimensional, the sequence of orbital excursions can be specified by a one-dimensional "return" map. To construct the map, we proceed as follows:

Figure 2. Poncaré section (left) constructed by slicing the attractor with a plane (red line) normal to the page (right). Points (left) on the section (green crosses, right) approximate a one-dimensional curve.

Figure 3. Unimodal return map computed from the Poincaré section.

1. Poincaré Section. First, we compute a Poincaré section by slicing the attractor as shown in Figure 2. Here (right side) the red line represents a plane normal to the page and the green crosses successive intersections of the asymptotic solution and the plane.

2. Parameterization. Fitting the intersections to a polynomial regression, we next assign to each point a distance, d_i, from one end of the section.

3. Display. Finally, we imagine that there exists a function, F(), such that d_i+1 = F(d_i), and plot i.e., d_i+1 vs. d_i for all pairs of points, i and i+1.

The surprising result (Figure 3) of this procedure is that the resulting return map is reminiscent of the logistic and other unimodal maps which may therefore be viewed as models of continuous chaos when the phase space is strongly contracting.⁽⁴⁾   A topologically equivalent construction (next amplitude map) is obtainable by plotting successive maxima (or minima), Max_i+1 vs. Max_i, in the time series, x(t), y(t) or z(t).

Stretching and Folding

The geometric basis of chaos is stretching and folding. The results are (1) trajectorial divergence, i.e., solutions based at nearby points separate, and (2) the identification of trajectories based at points distant from each other. Stretching and folding in the Rössler attractor can be visualized by viewing the evolution of the Poincaré section as the slicing plane is rotated through 360°. In the accompanying animation (click here), one can observe the section as it stretches out and then folds back down on itself. In effect, the nonlinear, dynamical baker rolls out the dough and folds it; rolls out the dough and folds it, ...

Phase Coherence

Figure 4. Power spectrum computed from {x(t)} for spiral chaos in Rössler's equations. The peaks are "instrumentally" sharp.

Stretching results in trajectorial divergence while folding causes mixing - two essential attributes of chaos which, in turn, give rise to "sensitivity to initial conditions." Sensitivity to initial conditions is what makes chaotic systems so special. Because one can never characterize a system's initial state to infinite precision, it follows that long-term chaotic evolution can never be predicted. ⁽⁵⁾   The best that one can do is to observe that there will be a probability distribution according to which neihborhoods of an attractor are vistited. Occasionally, this "invariant measure," can be calculated from a formula, ⁽⁶⁾   but, in general, this is not possible.

In the Rössler attractor, stretching and folding is largely confined to curves radiating outward from the saddle focus at the center of the disk. It follows that the motion is phase coherent, which is to say that there is very little azimuthal divergence, even though trajectories are thoroughly mixed radially. ⁽⁷⁾   Coherence in the phase space is reflected in the power spectrum (Figure 4) by the presence of instrumentally sharp peaks superposed on a noisy background. By "instrumentally sharp," we mean that the peaks are no broader than one expects from finite sampling.

The Turn of the Screw

Figure 5. Screw chaos in Rössler's equations. The "screws" result from the influence of a second saddle focus on trajectories rising out of the plane, z=0. Equations of motion as before; parameter values are a = 0.343; b = 1.82; c = 9.75).

Figure 6. Power spectrum computed from {x(t)} time series for the Rössler funnel. With the onset of screw chaos, the spectrum broadens. Compare with the corresponding spectrum for the simple band (sprial chaos) above.

Figure 7. Screw chaos in a three species (one predator, two prey) ecological model. The equations are parameterized to caricature the "keystone predator" scenario: In the absence of predators, prey species #1 outcomptes its rival; but prey species #1 is the prefered food source of the predators. Colors indicate trajectorial velocity as the system moves on the attractor.

Figure 8. Rössler-like chaos in a detailed model of the peroxidase-oxidase reaction. Shown here is a next-amplitude map computed by plotting successive maxima in the concentration of coIII, an exzyme intermediate, in temporal sequence. The noticable thickness reflects the fact that in this system, contraction down to an almost two-dimensional surface is not so pronounced.

Rössler coined the term "spiral chaos" to decribe the geometry induced by his equations. As it turns out, spiral chaos is the simplest form of chaos observable in continuous dynamical systems. Although the term has never been made mathematically precise, its essential attributes are those illustrated above:

Both spiral and screw chaos have been observed ⁽⁸⁾   in Lotka-Volterra models in which there is a keystone predator. ⁽⁹⁾   Here, species diversity at the victim trophic level is promoted by predators, the presence of which prevents one victim species from outcompeting another. Ecologists typically formulate the concept in terms of equilibria, but fixed point behavior is not necessary. The two forms of Rössler chaos are also observed in three-level food chain models ⁽¹⁰⁾   in which, for example, vegetation is consumed by herbivores and herbivores by carnivores. An example of screw chaos is shown in Figure 7. The colors indicate the velocity of the evolving trajectory (red signifies slow; violet, fast).

Chemical Oscilators.

One of the first experimental observations of chaos was reported by Roux et al.⁽¹¹⁾    for an oscillating chemical reaction. These authors used the so-called "method of delays"⁽¹²⁾    to generate three-dimensional trajectories from univariate time series. From the trajectories, they constructed unimodal return maps by taking Poincaré sections.

Spiral chaos also appears to be present in the peroxidase-oxidase reaction which is the simplest biochemical oscillator evidencing complex behavior. In this case, a detailed chemical model consisting of ten coupled differential equations gives good agreement with the experimental observations. ⁽¹³⁾   Figure 8 displays a unimodal return map induced by the model in the region of spiral chaos. The experimental results, while somewhat messier, are surprisingly similar. ⁽¹⁴⁾  

Infinite Dimensional Systems.

Both delay and partial differential equations (DDEs, PDEs) are equivalent to ordinary differential equations (ODEs) of infinite order and, thus, to systems consisting of an infinte number of ODEs. Even though the dimension of the phase space in these cases is infinite, low dimensional dynamics are often observed. Examples of Rössler-like chaos in infinite-dimensional systems can be found in the papers by Mackey and Glass ⁽¹⁵⁾   and Kuramoto. ⁽¹⁶⁾  

Exercises

1. Rössler's equations have two equilibria. Compute them and show that that one is a saddle focus of index 2 and the other a saddle focus of index 1. Recall that the index of a fixed point is the number of unstable directions, i.e., as reflected by the dimension of the unstable eigenspace.

2. Describe how one might visualize the stable and unstable manifolds of the equilibria numerically.

3. If you have access to a computer which can be programmed to do this sort of thing, implement your ideas in software and generate said visualization.