2008 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 5-6, 2008

Chaotic Oscillator Using Two RC Circuits
Shuhei MASUDA, Yumiko UCHITANI and Yoshifumi NISHIO
Dept. Electrical and Electronic Engineering,Tokushima University
Email: shuhei-614@ee.tokushima-u.ac.jp

Abstract— In this study, a simple chaotic circuit using two
coupled RC circuits are proposed. We have confirmed that thecircuit generates chaotic oscillation by both computer calculations
and circuit experiments.

I. I NTRODUCTION
There have been many simple circuits exhibiting chaotic oscillations. In particular, simple oscillators excited by periodic
signals can be a good circuit model to study the mechanismof chaos generation, because some of them can be analyzed
in a rigorous way. In [1], Tang et al. investigated a simple
multivibrator with periodic pulse and explained the generationof chaos using one-dimensional map. Torikai et al. proposed achaotic pulse generator realized by an oscillator and a periodic
signal and also controlled the generated chaos [2].
In this study, we propose a simple chaotic oscillator using
two RC circuits. In the circuit model, the two RC circuits arecoupled via comparators whose input is controlled by external
periodic signal. Computer simulations and circuit experiments
confirm the generation of chaos.
II. C IRCUIT MODEL
Figure 1 shows the circuit model. In the circuit, two RCcircuits are coupled via simple comparators of operational
amplifiers. The rectangular voltage wave is inputted to theother input terminals of the comparators and the comparators
produce the output voltage ±E which is their power supply
voltage according to the input signals. Figure 2 shows the1V2 2CVs

V1

By using the following variables and the parameters,
v1 = E x1 , v2 = E x2 , t = RC τ,(2)the normalized circuit equations are given as follows.
dx1= −x1 ± 1dtdx2
= −x2 ± 1dt

x1 = k1 e−τ ± 1,

V1x2 = k2 e−τ ± 1,

R3

Rectangular voltage wave.

(3)

Because the circuit equation is linear in each region, theexact solution can be derived as follows.

R3Fig. 2.

(4)1

where k1 and k2 are the arbitrary constants and ± are decided
according to the outputs of the comparators.

2CV2III. C OMPUTER CALCULATED RESULTS
Fig. 1.

Circuit model.

rectangular voltage wave inputted to the comparators. The
circuit equations are described as follows.
dV1
RC= −V1 ± E
dt
(1)
dV2
RC= −V2 ± E
dt

Figures 3(a) and (b) show computer calculated results for
V /E = 0.075 and T /RC = 3.0. We can confirm that thecircuit exhibits chaos.
IV. C IRCUIT EXPERIMENTAL RESULTS
Figures 4(a) and (b) show circuit experimental results for
C = 47µF and R = 1.2Ω. These results agree well to thecomputer calculated results.

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v2 2.5V

Att: x1-x20V
-0.4

0.4

-2.5V
-2.5V0V

2.5V

v1
(a)
-0.4

(a)

v1x1
0.3

0v2

-0.3

x2

t0.3

(b)

0Fig. 4. Computer simulation results. (a) Attractor x1 –x2 . (b) Time waveform.
-0.3

Vs

ACKNOWLEDGMENTS
0.3

This work was partly supported by Yazaki Memorial Foundation for Science and Technology.

0R EFERENCES
-0.3

(b)
Fig. 3. Computer simulation results. (a) Attractor x1 –x2 . (b) Time waveform.

[1] Y.S. Tang, A.I. Mees and L.O. Chua, “Synchronization and Chaos,”IEEE Trans. Circuits Syst., vol. 30, no. 9, pp. 620-626, Sep. 1983.
[2] H. Torikai, T. Saito and W. Schwarz, “A Chaotic Pulse Generator
and Sawtooth Control for Information Processing,” IEEE ISCAS 1997,
pp. 729-732, Jun. 1997.

V. C ONCLUSIONSIn this study, we have proposed a simple chaotic oscillator
using two RC circuits. Computer simulations and circuit
experiments confirmed the generation of chaos.

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