1976 The Rössler system

 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 .

(a) Folded chaos solution to the Rössler system. (b) First-return map to a Poincaré section. Fig. 1 : Spiral chaos solution to the Rössler system.

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].