Christophe LETELLIER
19/04/2020

During a stay in Nice (December 1991), **Hiroshi Kawakami** (Tokushima University, Japan) designed and **René Lozi** (University of Nice, France) an electronic circuit producing chaotic oscillations [1] It was presented at a conference organized at the Research Institute of Mathematical Science RIMS in Kyoto (18-21 February, 1992) by Shigehiro Ushiki. This circuit is of a particular interest because it has only two variables and a dynamical swtich. This is thus a two-dimensional piecewiese linear circuit. The dynamical nature of its switch allows to get chaos although its dimension is only two. This means that a full description of the switch would require at least a third variable.

The electronic circuit correspoding to the Alpazur is shown in Fig. 1. It has a switch and a nonlinear conductor. It consists in a capacitor *C*, an inductor *L* and a nonlinear cubic
resistance which can be negative. Two voltage sources *E*_{1} and *E*_{2} are alternatively connected to the circuit through the *dynamical switch* S. Compared to most of the piecewise linear circuit as the Chua circuit, this switch does not depend on the state but how the system reaches that state. Thus the source *E*_{1} is connected when the voltage *v* reaches the value -1 : it is then replaced with the source *E*_{2} when the voltage *v* returns to -0.1. The states of this circuit are
described by the voltage *v* on the oscillator and the current *i* through the inductor.

**Fig. 1. Block diagram for the electronic circuit of the Alpazur oscillator.**

Using Kirchoff’s laws, and then some change of variables, the dynamics of this circuit is governed by [2]

where parameters are [3]

With the parameter values

the Alpazur oscillator produces a chaotic attractor which is characterized by a tri-modal map (Fig. 2).

**Fig. 2. Chaotic attractor produced by the Alpazur oscillator.**

There is a period-doubling cascade issued from a period-1 window leading to this attractor (Fig. 3).

**Fig. 3. Bifurcation diagram of the Alpazur oscillator.**

This oscillator took its name from the Hotel where Hiroshi spent his stay in Nice (Fig. 4).

**Fig. 4. Hotel Alp’Azur, Nice, France**

[1] **H. Kawakami & R. Lozi**, Switched dynamical systems. Dynamical of a class of circuits with switch-Structure and bifurcations of dynamical systems, in Proceeding of the RIMS Conference, *Advances Series in Dynamical Systems*, **11**, 39-58, World Scientific (Singapore), 1992. Online

[2] **T. Kousaka, T. Ueta & H. Kawakami**, Bifurcation of switched nonlinear dynamical systems. *IEEE Transactions on Circuits and Systems II : Analog and Digital Signal Processing*, **46** (7), 878-885, 1999. Online

[3] **D. Chirita, V. Stefanescu, B. C. Florea & A. A. Stoichescu**,
An analogical control method of the Alpazur oscillator, *Proceedings of the 9th Int. Symposium on Advanced Topics in Electrical Engineering*, (May 7-9, 2015
Bucharest, Romania), 854-857, 2015. Online