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1976 The Rössler system

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] :

 
 \left\{
      \begin{array}{l}
         \dot{x} = -y-z \\
         \dot{y} = x+ay \\
         \dot{z} = b+z(x-c) 
  \end{array}
  \right.

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 a=0.420, b=2 and c=4.

(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].

PNG - 8 ko
Fig. 2 : The "three-dimensional blender".
\rightarrow = 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. \uparrow = "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.

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