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1992 The Alpazur oscillator

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 E1 and E2 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 E1 is connected when the voltage v reaches the value -1 : it is then replaced with the source E2 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.

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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]

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where parameters are [3]

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With the parameter values

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

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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).

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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).

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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

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