2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012

Coupled Chaotic Oscillators Composed of RC Circuits
Ayana Shimada† , Hironori Kumeno† , Yoko Uwate† , Yoshifumi Nishio† and Jingmin Xin‡
† Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Phone: +81-88-656-7470
Email: {ayana,kumeno,uwate,nishio}@ee.tokushima-u.ac.jp‡ Xi’an Jiaotong University
28 Xianning Xilu, Xi’an, Shaanxi 710049, China
Phone: +86-29-8266-8802
Email: jxin@mail.xjtu.edu.cn

Abstract
In this study, we propose coupled chaotic oscillators composed of RC circuits. We carry out computer simulations andinvestigate chaotic behavior when the number of RC circuits
are increased.

1. Introduction
Recently, many researchers have shown their interests inchaotic systems. Chaos has been investigated not only inengineering but also in various fields such as medicine, sociology and economics. In the field of electrical and electronic engineering, researchers have proposed various applications using chaos as chaos communications, chaos cryptosystem, chaos neural networks and so on. In order to realize chaotic engineering systems, it is important to investigate simple chaos-generating circuits. In our previous study,a simple chaotic oscillator using two RC circuits were proposed [1].
In this study, we propose coupled chaotic oscillators composed of RC circuits. By computer simulations, we investigate the chaotic behavior when the number of the coupled RCcircuits are increased.

2. Circuit model

Figure 1: Circuit model.
The circuit equations are described as follows:{−v1 − E (v2 > VS )dv1RC=
dt−v1 + E (v2 < VS ){−v2 + E (v1 > VS )
dv2=
RC
dt
−v2 − E (v1 < VS ){−vn − E (vn+1 > vn−1 )
dvn
RC=
dt
−vn + E (vn+1 < vn−1 ){−vn+1 + E (vn > vn−1 )dvn+1RC=dt−vn+1 − E (vn < vn−1 )Figure 1 shows the circuit model. In this figure 15 chaotic
oscillators, which was proposed in [1], are coupled via oneway coupling through simple comparators. An independent
rectangular voltage source is connected only to the first oscillator and the voltage across one of the two capacitors in theprevious stage of oscillators is inputted to the comparators in(1)the next oscillator. Hence, the whole circuit consists of one
rectangular voltage source, 30 comparators, 30 resistors, and(n = 3, 5, 7, 9, ... , 29)
30 capacitors. Figure 2 shows the rectangular voltage waveform inputted to the first oscillator. VS is the amplitude of the where E is the output voltage of the comparators, namely theDC supply voltage of the operational amplifiers. By using therectangular voltage and T is the period of the waveform.

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the average values of the standard deviation of the points fordifferent α. We can see that the dispersion becomes larger as
α increases.
Finally, we observe chaotic attractors and the Poincarémaps for different α values. Figure 8 shows chaotic attractors and Fig. 9 shows their corresponding Poincaré maps. As
α increases, the size of the attractors clearly becomes large.
However, only from these figures we cannot say whether if
the complexity becomes large or small. Hence, further investigations are needed.

Figure 2: Rectangular voltage wave.
following variables and the parameters,(1)
vk = Exk , t = RCτ, VS = Eα, T = RCβ,(2)(k = 1, 2, ... , 30)
we obtain the normalized circuit equations. Because the circuit equations are linear in each region, the rigorous solutionof the circuit equations can be derived as follows:{(x10 + 1)e−τ − 1 (x2 > α)x1 =(x10 − 1)e−τ + 1 (x2 < α){(x20 − 1)e−τ + 1 (x1 > α)x2 =(x20 + 1)e−τ − 1 (x1 < α){(xn0 + 1)e−τ − 1 (xn+1 > xn−1 )xn =(xn0 − 1)e−τ + 1 (xn+1 < xn−1 ){(x(n+1)0 − 1)e−τ + 1 (xn > xn−1 )(3)
xn+1 =(x(n+1)0 + 1)e−τ − 1 (xn < xn−1 )

(2)

(3)

(4)

(5)

(n = 3, 5, 7, 9, ... , 29)
where α is a parameter corresponding to VS , x10 , x20 and xn0
are initial values (n = 3, 5, 7, 9, 11, ... , 29).
3. Simulation result(6)

First, Fig. 3 and Fig. 4 show the attractors and their corresponding Poincaré maps, respectively, obtained from somechaotic oscillators. Comparing these figures, we can observe
more complex attractors as the positions of the circuits increase.
Next, Figs. 5 and 6 are time waveforms. It is interestingthat we can find essential difference between these two figures. Namely, in Fig. 5 chaotic waveforms are completely
unsynchronized while in Fig. 6 chaotic waveforms of x2 -x29
seem to almost synchronize in in-phase. The detailed analysisof these synchronization phenomenon is our important futureresearch.
In order to evaluate chaotic features, we investigate the dispersion of the points on the Poincaré maps. Figure 7 shows

(a)

(b)

Figure 3: Chaotic attractor obtained from computer simulation (β = 4.0). (a) α = 0.097 and (b) α = 0.180. (1) x1 -x2 ,(2) x5 -x6 , (3) x11 -v12 , (4) x17 -x18 , (5) x23 -v24 and (6) v29 v30 .

4. Conclusions
In this study, we have proposed coupled chaotic oscillatorscomposed of RC circuits. We carried out computer simula-- 18 -(1)

(2)

(3)

(4)(5)
Figure 5: Time waveforms for α = 0.097 and β = 4.0. From
top x1 , x3 , x5 , x7 , x9 , x11 , x13 , x15 , x17 , x19 , x21 , x23 , x25 ,x27 , x29 .
(6)

(a)

(b)

Figure 4: Poincaré map obtained from computer simulation(β = 4.0). (a) α = 0.097 and (b) α = 0.180. (1) x1 -x2 ,(2) x5 -x6 , (3) x11 -v12 , (4) x17 -x18 , (5) x23 -v24 and (6) v29 v30 .

tions and investigated chaotic behavior when the number of
RC circuits were increased.
One of our future works is to carry out circuit experiments.
References[1] S. Masuda, Y. Uchitani and Y. Nishio, “Simple Chaotic
Oscillator Using Two RC Circuits,” Proc. of NCSP’09,
pp. 89-92, 2009.

Figure 6: Time waveforms for α = 0.180 and β = 4.0. From
top x1 , x3 , x5 , x7 , x9 , x11 , x13 , x15 , x17 , x19 , x21 , x23 , x25 ,x27 , x29 .

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(1)Figure 7: Change of standard deviation for different α. Onlyx5 , x11 , x17 , x23 and x29 are shown.
(2)

(3)

(1)

(4)

(2)

(5)
(a)

(3)

(b)

(c)

(d)

(e)

Figure 9: Poincaré maps obtained from computer simulation.
(β = 4.0). (a) α = 0.027, (b) α = 0.040, (c) α = 0.061,(d) α = 0.082, and (e) α = 0.097. (1) x5 -x6 , (2) x11 -x12 ,(3) x17 -x18 , (4) x23 -x24 , and (5) x29 -x30 .

(4)

(5)
(a)

(b)

(c)

(d)

(e)

Figure 8: Chaotic attractors obtained from computer simulation (β = 4.0). (a) α = 0.027, (b) α = 0.040, (c) α = 0.061,(d) α = 0.082, and (e) α = 0.097. (1) x5 -x6 , (2) x11 -x12 ,(3) x17 -x18 , (4) x23 -x24 , and (5) x29 -x30 .

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