Christophe LETELLIER
01/05/2022

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

where the nonlinear switch is the piecewise linear function

With appropriate parameter values as

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

**Fig. 1. Chaotic attractor produced by the modified Wien-bridge oscillator.**

Using the Poincaré section

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.

**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 C^{1}T^{1} 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.