1978 Torn Unimodal Chaos

Christophe LETELLIER
Otto E. Rössler & Peter J. Ortoleva

- What torn unimodal chaos is ?

It is known that for producing a chaotic behavior, sensitivity to initial conditions is combined to some recurrence properties. These two specific characteristics result from two mechanisms : stretching and squeezing. This can be produced by a folding or a tearing. Typically, an attractor involving a folding is produced by the Rössler system and one involving a tearing is the Lorenz system. These two mechanisms were investigated in [1]. Torn Unimodal chaos corresponds to an attractor with a tearing mechanism that is characterized by a cusp --- or a Lorenz --- map. The Lorenz system is a good example but it has a rotation symmetry. The purpose here is to have an attractor with a tearing mechanism without any symmetry.

- The system

To the best of our knowledge, the first set of equations that was identified to produce a chaotic attractor bounded by a genus-1 torus and possessing a Lorenz map was proposed by Otto Rössler and Peter Ortoleva from Indiana University [2] as an isothermal abstract reaction system. The systems reads :

      \dot{x} = ax+by-cxy -\frac{(dz+e)x}{x+K_1} \\[0.4cm]
      \dot{y} = f+gz-hy-\frac{jxy}{y+K_2} \\[-0.4cm]
      \displaystyle \dot{z} = k+lxz-mz

This abstract chemical reaction produces a torn unimodal chaotic attractor as shown in Fig. 1. Parameter values are a=33, b=150, c=1, d=3.5, e=4815, f=410, g=0.59, h=4, j=2.5, k=2.5, l=5.29, m=750, K1=0.01 and K2=0.01. A first-return map to a Poincaré section (Fig. 2) has the shape of the Lorenz map as expected. The l-value is slightly modified to obtain a Lorenz map without a gap between the two monotonic branches as originally published [2].

PNG - 58.1 ko
Fig. 1 : Chaotic attractor with a tearing mechanism
PNG - 14.3 ko
Fig. 2 : Lorenz map with its characteristic cusp

[1] G. Byrne, R. Gilmore & C. Letellier, Distinguishing between folding and tearing mechanisms in strange attractors, Physical Review E, 70, 056214, 2004.

[2] O. E. Rössler & P. J. Ortoleva, Strange attractors in 3-variable reaction systems, Lecture Notes in Biomathematics, 21, 67-73, 1978.

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