1995 A Wien-bridge oscillator

Christophe LETELLIER

In the 1980s, many electronic circuits producing chaotic regimes were proposed. Most of the proposed circuits are modifications of a LC circuit which is not convenient to produce audio and lower frequencies. The RC circuits are more acceptable for low frequencies but they are often rather complex. A. Namajhas and Arunas Tamasevicius proposed a simple electronic circuit made of a Wien bridge, a single opamp and a single nonlinear device displaying a current saturation characteristic [1]. The equations governing this electronic circuit are

PNG - 3.4 ko

where the nonlinear switch is the piecewise linear function

PNG - 2.2 ko

With appropriate parameter values as

PNG - 2 ko

a chaotic attractor is obtained as plotted in Fig. 1.

PNG - 7.4 ko
Fig. 1. Chaotic attractor produced by the modified Wien-bridge oscillator.

Using the Poincaré section

PNG - 2 ko

a first-return map is computed (Fig. 2) : it is a unimocal map which is associated with a complete symbolic dynamics, as indicated by the increasing branch with touches the first bisector.

PNG - 2.4 ko
Fig. 2. First-return map to a Poincaré section of the chaotic attractor produced by the Wien-bridge oscillator.

According to the recent taxonomy of chaos [2], this unimodal chaotic attractor is a C1T1 bounded by a genus-1 torus and characterized by a unimodal smooth map with a global torsion of a half-turn.

[1] A. Namajhas & A. Tamaievitius Modified Wien-bridge oscillator for chaos. Electronic Letters, 31 (5), 335–336, 1995.

[2] C. Letellier, N. Stankevich & O. E. Rössler, Dynamical Taxonomy : some taxonomic ranks to systematically classify every chaotic attractor International Journal of Bifurcation & Chaos, 32 (2), 2230004, 2022.

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