Christophe LETELLIER
11/01/2008

Otto E. Rössler

**The system**

The most popular example of the a simple folding remains the Rössler system proposed in 1976 [1] :

It is characterized by a first-return map with a differentiable critical point separating the increasing and decreasing branches. The Rössler attractor is observed with parameter values , and .

Rössler called this simple stretched and
folded ribbon, the *spiral* chaos. He already sketch the structure of the
attractor with a “*paper model*” as shown in Fig. 2 [2].

**Fig. 2 : The "three-dimensional blender".**- = trajectories entering the structure from the outside ; 1, 2 = half cross-section (demonstrating the "mixing transformation" that occurs), e = entry point of some arbitrarily chosen trajectory, r = reentry point of the same trajectory after one cycle. = "horseshoe map", a.sl. = allowed slit.

For other parameter values, a more developed chaos is produced. The first-return map to a Poincaré section has now many monotonic branches
(Fig. 3). Rössler named that type of chaos, the “*funnel chaos*”. Its topology has been investigated by Letellier *et al* [3]. The spiral chaotic attractor corresponds to a phase coherent attractor while the funnel type is phase incoherent [4].

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

[2] **O. E. Rössler**, Chaotic behavior in simple reaction system, *Zeitscrift für Naturforschung A*, **31** (3-4), 259-264, 1976.

[3] **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.

[4] **J. D. Farmer, J. P. Crutchfield, H. Fröling, N. H. Packard & R. S.
Shaw**, Power spectra and mixing properties of strange attractors.
*Annals of the New York Academy of Sciences,* **357**, 453-472, 1980.