2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009

Simple Chaotic Oscillator Using Two RC Circuits
Shuhei MASUDA,Yumiko UCHITANI and Yoshifumi NISHIO
Tokushima University
2-1 Minami-Josanjima-cho,Tokushima 770-8506, JAPAN
Phone/FAX:+81-88-656-7470
Email: shuhei-614@ee.tokushima-u.ac.jp

Vs (t)

Abstract
In this study, a simple chaotic circuit using two coupled RCcircuits is proposed. We confirm that the circuit generates
chaotic oscillation by both computer calculations and circuit
experiments.

+V

1. Introduction
There have been many simple circuits exhibiting chaoticoscillations. In particular, simple oscillators excited by periodic signals can be a good circuit model to study the mechanism of chaos generation, because some of them can be analyzed in a rigorous way. In [1], Tang et al. investigateda simple multivibrator with periodic pulse and explained thegeneration of chaos using one-dimensional map. Torikai etal. proposed a chaotic 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
are coupled via comparators whose inputs are controlled byexternal periodic signal. Computer simulations and circuit
experiments confirm the generation of chaos.
2. Circuit model+E

V2
V s (t )V1

C+E

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 the rectangular voltage wave inputted to thecomparators. V is the amplitude of the rectangular voltage. Tis the period of the rectangular voltage. The circuit equations
are described as follows.
When V1 > Vs (t) and V2 < Vs (t)dV1 RC dt = −V1 + E,
dV2
RC= −V2 + E,dt

V1R+−CT

Figure 2: Rectangular voltage wave.

R+−−E−VV2(1)When V1 < Vs (t) and V2 < Vs (t)−E
dV1
 RC dt = −V1 + E,Figure 1: Circuit model.

 RC dV2 = −V2 − E,dt- 89 -(2)When V1 > Vs

and V2 > Vs


dV1

 RC dt = −V1 − E,

where α(τ ) is a function corresponding to Vs (t), k1 and k2
are the arbitrary constants determined by initial values. We(3) named four linear regions as Dpp , Dpm , Dmp and Dmm.


 RC dV2 = −V2 + E,dtWhen V1 < Vs (t) and V2 > Vs (t)3. Generation of chaos
Figures 3 (a) and (b) show computer calculated results for
α = 0.075 and β = 3.0. We can confirm that the circuit exhibitschaos.

dV1


 RC dt = −V1 − E, RC dV2 = −V2 − E,dt
By using the following variables and the parameters,
v1 = E x1 , v2 = E x2 , t = RC τ,
V = Eα, T = RCβ,the normalized circuit equations are given as follows.
dx1
 dt = −x1 ± 1,
 dx2
= −x2 ± 1.
dt(4)

X2
0.4

(5)

-0.4

(6)

0.4

X1

Because the circuit equations are linear in each region, theexact solutions can be derived as follows.
When x1 > αs (τ ) and x2 < αs (τ ) (Dpp),(-0.4
x1 = k1 e−τ+ 1,x2 = k2 e

−τ

+ 1,

x1 = k1 e

0.3

x1

When x1 < αs (τ ) and x2 < αs (τ )
((a)

(7)

−τ

(Dpm),-0.3

x2+ 1,0.3(8)x2 = k2 e−τ − 1,-0.3

αWhen x1 > αs (τ ) and x2 > αs (τ )(0.3(Dmp),

x1 = k1 e−τ − 1,-0.3(9)x2 = k2 e−τ + 1,

τ(b)When x1 < αs (τ ) and x2 > αs (τ )(x1 = k1 e−τ − 1,x2 = k2 e−τ − 1,

Figure 3: Computer simulation results. (a) Attractor on x1 –x2 plane. (b) Time waveform.

(Dmm),(10)- 90 -

Figures 4 (a) and (b) show circuit experimental results for
C = 47µF , R = 1.2Ω and f = 6.75kHz. These results agree
well to the computer calculated results.

v2 2.5V
α=0.65α=0.60

α=0.25α=0.24α=0.11α=0.075

0V-2.5V-2.5V

0V

2.5V

v1

Figure 5: Bifurcation of chaos attractor.

(a)Figure 8 shows the switching points from other three linear
regions to one region (α = 0.075 and β = 3.0). The points
are dotted only when the region switches to one region fromother three regions. We can see that (a) and (d) are origin
symmetry, (b) and (d) are too. The distance between two lines
shown in four regions is always 2α.

v1v2

5. Conclusionst(b)Figure 4: Circuit experimental results. (a) Attractor on v1 –v2plane. (b) Time waveform.
4. Analysis of chaotic attractor

In this study, we have proposed a simple chaotic oscillator using two RC circuits. Computer simulations and circuit
experiments confirmed the generation of chaos. In the future
work, we will advance theoretical analysis of this circuit andtry to consider engineering application.
Acknowledgments

This work was partly supported by Yazaki Memorial FoundaFigure 5 shows bifurcation of the attractor when β is held at tion for Science and Technology.
3.0 and α changes from 0.65 to 0.075. We can confirm changing process of attractors. At first, the attractor shows that twoReferences
waves x1 and x2 are synchronized at anti-phase. However, as
α is smaller, synchronization begins to break. And finally, the[1] Y.S. Tang, A.I. Mees and L.O. Chua, “Synchronization
attractor shows chaotic behavior.
and Chaos,” IEEE Trans. Circuits Syst., vol. 30, no. 9,
Figure 6 shows only the part in each region of the chaotic
pp. 620-626, Sep. 1983.
attractor.
We can know convergence points in four linear regions [2] H. Torikai, T. Saito and W. Schwarz, “A Chaotic Pulse
from equations (7) − (10). Figure 7 shows convergence
Generator and Sawtooth Control for Information Propoints and formulas of the most outward line in four linear
cessing,” Proc. of ISCAS’97, pp. 729-732, Jun. 1997.
regions (α = 0.075 and β = 3.0). From this result, we
can confirm the most outward lines in Dpp and in Dmm areparallel, also in Dpm and in Dmp are parallel.

- 91 -DppDpp( x1,x2) = (1,1)

Dpm

x2=1.37x1-0.37
x2=-0.66x1-0.34( x1,x2) = (1,-1)(a) Dpp(b) Dpm

Dmp( x1,x2) = (-1,1)

Dmm(a) Dppx2=0.66x1+0.34

x2=-1.37x1+0.37

Dpm( x1,x2) = (-1,-1)(c) Dmp(d) DmmFigure 7: Convergence points and formulas of the most outward line in four linear regions. (α = 0.075 and β = 3.0.)(b) Dpm
Dmp
Dpp

Dpm

2α2α(a) Dpp(b) Dpm(c) Dmp
Dmp

Dmm

Dmm
2α2α(c) Dmp(d) DmmFigure 8: Switching points. (α = 0.075 and β = 3.0.)(d) DmmFigure 6: Four linear regions. (α = 0.075 and β = 3.0.)- 92 -