1995 A Rössler-like oscillator
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.](local/cache-TeX/6fd10d4d9c0993fe663163edc5b83ae7.png "\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.](local/cache-TeX/403c6dd3ba30d9edb0b18b997c87cbc5.png "G(x) =
\left|
\begin{array}{ll}
0 & \mbox{ if } x \leq 3 \\[0.1cm]
\mu (x-3) & \mbox{ if } x > 3
\end{array}
\right.")
The electronic circuit corresponding to these equations is shown in Fig. 1. The three variables correspond to three voltages measured in the circuit as indicated in Fig. 1.
- Fig. 1. Electronic circuit. All OP amps are of type 741.
Using parameter values as used in [3], that is, as
- 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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MàJ . 04/02/2024