Rossler attractor

Figure 1: Stereoscopic view of the Rössler attractor. Parameter values: a=0.432, b=2 and c=4.

The Rössler attractor is a chaotic attractor solution to the system \[ \dot{x} = -y -z \] \[ \dot{y} = x+ay \] \[ \dot{z} = b+ z(x-c) \] proposed by Rössler (1976), often called Rössler system. Here, \((x, y, z) \in \mathbb{R}^3\) are dynamical variables defining the phase space and \((a, b, c) \in \mathbb{R}^3\) are parameters.

[edit] Properties

[edit] Fixed points

Rössler system has two fixed points (often called equilibria)\(F_\pm\) located at \[ (x_\pm,y_\pm,z_\pm)=\left( \frac{c \pm \sqrt{c^2-4ab}}{2},-\frac{c \pm \sqrt{c^2-4ab}}{2a},\frac{c \pm \sqrt{c^2-4ab}}{2a} \right) \]

Figure 2: Schematic view of the fixed points and their manifolds.

A fixed point \(F_-\) is located in the middle of the attractor and is a saddle-focus with an unstable 2D manifold - an unstable spiral mainly in the \(x-y\) plane - when the trajectory settles down onto a chaotic attractor. \(F_+\) is outside of the region of the attractor. The nonlinearity \(z(x-c)\) becomes active when the trajectory leaves the \(x-y\) plane. The trajectory thus visits the neighborhood of \(F_+\) - also a saddle-focus - whose 1D unstable manifold sends the trajectory along the 1D stable manifold of \(F_-\ .\) A new cycle can then occur. With appropriate parameter values, the trajectory thus describes a chaotic attractor.

[edit] Topological description

Figure 3: The Rössler attractor in the 3-d space. The viewpoint is rotating around the attractor. Parameter values: a=0.432, b=2 and c=4.

Figure 4: Paper-sheet model (exaggerated) showing the "normal band" and the "Möbius band".

The topology of the Rössler attractor was first described in terms of a paper-sheet model. Typically, the paper-sheet model can be divided in two stripes, one being a "normal band" and one being a Möbius band. These two bands thus define two different topological domains. In the 1990's, such a topological description led to the concept of a branched manifold, also called knot-holder or template (see Gilmore & Lefranc, 2002). It can be shown that such a paper-sheet model encodes all topological properties of the unstable periodic orbits embedded within the attractor. The Rössler attractor is the most simple chaotic attractor from the topological point of view, that is, it is a simple stretched and folded ribbon.

Figure 5: First-return map of the Rössler attractor. Parameter values: a=0.432, b=2 and c=4.

A Poincaré section of the Rössler attractor is conveniently defined as

\[ P \equiv \left\{ (y_n,z_n) \in \mathbb{R} ~|~ x_n=x_-,\dot{x}_n > 0 \right\} \, . \]

\((y_n,z_n)\) compose a very thin curve on the plane \(x=x_-\ ,\) and this curve is created by stretching and folding processes of the attractor. Such stretching and folding processes can be observed by taking the return map.

A first-return map of \(y_n\) to this Poincaré section is a simple parabola for the parameter values used for the figure shown above. This is therefore a unimodal map (i.e., with a single extremum) made of two monotonic branches separated by one maximum defining the boundary between the normal band - the increasing branch - and the Möbius band - the decreasing branch. The Rössler attractor can therefore be viewed as the trivial suspension - given a discrete map f of an n-dimensional manifold M, it is always possible to construct a flow on an n+1-dimensional manifold - of the Logistic map (both obey the Sharkovsky Ordering).

[edit] Bifurcation diagram

Figure 6: Bifurcation diagram versus parameter a of the Rössler system. Other parameter values: b=2 and c=4.

When a parameter value is varied, bifurcations may occur. In fact, with \( a \in [0.126~;0.43295] \ ,\) \( b=2 \) and \( c=4 \ ,\) there is (nearly) a one-to-one correspondence between the bifurcation diagram of the Rössler system and that of the Logistic map with \( \mu \in [1~;4] \ .\) Thus, the chaotic regime arises after a period-doubling cascade as in the Logistic map.

When \( a \) is greater than 0.43295, a second extremum appears as well as a third band in the paper-sheet model, leading to multimodal chaos, originally called "screw-type chaos" by Rössler. A complete topological description can be found in Letellier et al (1995).

In fact, there is not a single Rössler system but a full collection of different sets of ordinary differential equations with different topologies. Some references where these models can be found are given below.

[edit] Lyapunov spectra

Figure 7: Lyapunov spectra versus parameter a of the Rössler system. Other parameter values: b=2 and c=4.

When the chaotic attractor is stable, three Lyapunov spectra \(\lambda_1\ ,\) \(\lambda_2\ ,\) and \(\lambda_3\) (\(\lambda_1 \geq \lambda_2 \geq \lambda_3\)) satisfy \(\lambda_1>0\ ,\) \(\lambda_2=0\ ,\) and \(\lambda_3<0\ .\) On average, \(\lambda_1\) is the expantion rate of stretching process of the attractor, and \(\lambda_3\) is the reduction rate of the folding process. Because the absolute value of \(\lambda_3\) is much larger than that of \(\lambda_1\ ,\) the normal band and the Möbius band in the attractor become very thin.

[edit] Reminiscences about the discovery of the system

Otto E. Rössler in 2006: "I am a very visual person, I apologize for that. Curves in 3D-space fascinate me. I had planned to generate a "knotted limit cycle" when Art Winfree told me about the existence of chaos in 1975; so my mind was flooded by the beauty of the Lorenz attractor when Art introduced me to it asking me to set up a chemical reaction system with the same behavior. When unsuccessful (only later Peter Ortoleva and I succeeded jointly), I decided to settle for less. I wondered if a rope around the nose, circling it in several loops before falling off at the tip and then curving back to the starting point or its neighborhood, would not produce a similar tangle in 3D-space. Then this narrowing tunnel-like slinky got magically flattened in my mind into a spiral which strangely was expanding rather than contracting before being bent into a reinjection loop toward the neighborhood of its origin. (The opposite feat, reinjection inside-out, was later discovered by Normann Kleiner and Sebastian Fischer. This was much as Shilnikov had done topologically some ten years earlier in the other time direction, as I later learned.) Thus a letter-Z like slow manifold (in Christopher Zeeman's terms), but laterally extended into a sheet, was kind enough to offer itself as a host in my mind. One could prove the existence of chaos, with a 1D return map. But the equations, later re-discovered by Christian Mira, were messy. Floris Takens had a similar idea at about the same time. Simulating the expanding spiral on the lower floor of the letter-Z paper worked in December of 1975. Leaving the safe ground of singular perturbation was facilitated by remembering the rope around the nose. "Dirty" equations that ran faster on the slow desktop HP 9820A thereby became appealing in their own right. Stepwise simplification of the original singular-perturbation equations, with eventual omission of the upper knee in the letter Z, appeared legitimate. Robert Rosen first saw the 'spiral-chaos' simplified attractor slowly take shape on the slow desktop plotter with the built-in close to machine-language computer, and loved this illustration of his analogous-systems theorem (a long line), as he said, in his published paper in the Bulletin of Mathematical Biophysics in 1968.

The expanding-spiral component is a linear harmonic oscillator with negative damping:

\[ \dot{x} = -y \] \[ \dot{y} = x+0.2y \, . \] The threshold for \(z\)-variable, \(x=5.7\ ,\) with a constant influx of 0.2,

\[ \dot{z} = 0.2+z(x-5.7) \] sufficed if \(z\) was fed back into the first line: \[ \dot{x} = -y -z \] Later René Thomas saw much deeper into the topology of this feedback circuit. Hyperchaos became a natural successor in 4D. Still, the "re-injected rope" in 3D space still makes my heart jump when I'm in the right mood. It has a tender tickling connotation, when the rope comes back to your nose, but never quite to the same point. Imagine: a transfinitely exact and infinitely inventive angel is gently stroking your nose. The pleasurable laughter never stops."

[edit] References

R. Gilmore & M. Lefranc, The topology of chaos, Wiley, 2002.

C. Letellier, P. Dutertre & B. Maheu, Unstable periodic orbits and templates of the Rössler system: toward a systematic topological characterization, Chaos, 5 (1), 271-282, 1995.

C. Letellier, E. Roulin & O. E. Rössler, Inequivalent topologies of chaos in simple equations, Chaos, Solitons & Fractals, 28, 337-360, 2006.

R. Rosen, Turing's morphogens, two-factor systems and active transport, Bulletin of Mathematical Biophysics, 30, 493-499, 1968.

O. E. Rössler, An equation for continuous chaos, Physics Letters A, 57 (5), 397-398, 1976.

O. E. Rössler, Chaotic behavior in simple reaction system, Zeitschrift für Naturforsch A, 31, 259-264, 1976.

O. E. Rössler, Different types of chaos in two simple differential equations, Zeitschrift für Naturforsch A, 31, 1664-1670, 1976.

O. E. Rössler, Chaos in abstract kinetics: two prototypes, Bulletin of Mathematical Biology, 9, 275-289, 1977.

O. E. Rössler, Continuous chaos, in Synergetics (Proceedings of an International Workshop on Synergetics at Schloss Elmau, Bavaria (May 2-7, 1997), Ed. H. Haken), Springer-Verlag, 1977.

O. E. Rössler, Continuous chaos: four prototype equations, Annals of the New York Academy of Sciences, 316, 376-392, 1979.

Internal references

John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.