1979 The hyperchaotic Rössler system

Christophe LETELLIER
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- The system

Up until now we have studied chaotic behavior that is generated by the complementarity between a stretching direction and a squeezing direction. When the evolution of a system occurs in a four-dimensional space, a second stretching direction can appear. Sketch of hyperchaotic trajectory located at the intersection of two expanding directions S’ and one contracting surface S. A double difficulty is encountered when a description is attempted : one is related to the phase space which has four dimensions and another results from the layered structure due the superposition of the two stretching directions.(Fig. 1).

In 1979 Otto Rössler proposed the set of equations

      \dot{x} = -y -z \\
      \dot{y} = x + 0.25 y + w \\
      \dot{z} = 3.0 + xz \\
      \dot{w} = -0.5 z + 0.05 w

which produces such behavior.

Without the third variable z, the solutions to this system are ejected to infinity. Variable z acts as a retroaction beyond a given threshold and ensure the folding which sends the trajectory back to the center of the attractor (Fig. 2). Rössler called the behavior produced by this set of equations a hyperchaotic attractor [1]. A possible set of initial conditions is (x,y,z,w)=(-10, -6, 0, 10.1).

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Fig. 2. Hyperchaotic attractor produced by the four-dimensional Rössler system.

The equations in this model are not particularly complicated, other than for the fact that they are four-dimensional. The two stretching directions produce a first return map without any very clear structure ; this does not permit a partition to be constructed for this attractor (Fig. 3). Because of the thickness of the first return map it is quite unthinkable that a branched manifold can correctly describe the structure of this attractor. The topological description of such a structure is currently an open question...

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Fig. 3. First-return map to a Poincaré section of the hyperchaotic attractor.
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[1] O. E. Rössler, An equation for hyperchaos, Physics Letters A, 71, 155-157, 1979.

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