1995 A Rössler-like oscillator

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.

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

\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.

**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.

ATOMOSYD {2007} | | | | MàJ . 01/05/2022