telech

1995 A Rössler-like oscillator

Christophe LETELLIER
28/03/2016

In 1995, Thomas Carroll designed an easy-to-build electronic circuit for producing chaotic behaviors. He started from the Rössler equations [1] where he replaced the nonlinear term by a piecewise linear function [2], leading to the system


  \left\{
    \begin{array}{l}
       \dot{x}=-\alpha_x (x+\beta y +\Gamma z) \\[0.1cm]
       \dot{y}=-\alpha_y (-\gamma x + (1-\delta) y) \\[0.1cm]
       \dot{z}=-\alpha_z (-G(x) +z)
    \end{array}
  \right.

where the nonlinearity is the piecewise linear function

 G(x) = 
  \left|
    \begin{array}{ll}
       0 & \mbox{ if } x \leq 3 \\[0.1cm]
       \mu (x-3) & \mbox{ if } x > 3
    \end{array}
  \right.

Using parameter values as used in [3], that is, as


  \left\{
   \begin{array}{l}
      \alpha_x = 500 \\
      \alpha_y = 200 \\
      \alpha_z =10000 \\
      \beta=10 \\
      \Gamma = 20 \\
      \gamma = 50 \\
      \delta = 8.772 \\
      \mu = 15
  \end{array}
  \right.

JPG - 22.2 ko
Fig. 1. Chaotic attractor produced by the Rössler-like oscillator.

[1] O. E. Rössler, The Chaotic Hierarchy, Zeitschrift für Naturforschung A. 38 (7), 788-801, 1983.

[2] T. L. Carroll, A simple circuit for demonstrating regular and synchronized chaos, American Journal of Physics, 63 (4), 377-379, 1995.

[3] R. Sevilla-Escoboza, R. Gutiérrez, G. Huerta-Cuellar, S. Boccaletti, J. Gomez-Gardenes, A. Arenas,7 and J. M. Buldu, Enhancing the stability of the synchronization of multivariable coupled oscillators, Physical Review E, 92, 032804, 2015.

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