2013 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Synchronization Phenomena of Two Simple RC Chaotic Circuits Coupled by a Capacitor
Ayana Shimada1 , Yoko Uwate1 , Yoshifumi Nishio1 and Jingmin Xin2
1Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone: +81-88-656-7470
E-mail: {ayana,uwate,nishio}@ee.tokushima-u.ac.jp

2

Xi’an Jiaotong University
28 Xianning Xilu, Xi’an, Shaanxi, China
Phone: +86-29-8266-8802
Email: jxin@mail.xjtu.edu.cnAbstract

In this study, we investigate two coupled chaotic oscillatorscomposed of RC circuits. Two chaotic oscillators coupled by
In this study, we investigate two coupled chaotic oscillators one capacitor. We carry out computer calculation and circuitcomposed of RC circuits. We carry out computer calculation experiment about the cases of VS 1 = VS 2 and VS 1 = −VS 2 .
and circuit experiment. We could observe in-phase synchronization and anti-phase synchronization by computer calcu2. Circuit Modellation. Moreover, when we changed the value of couplingstrength between oscillators, we observed the change of synchronization state. In addition, we observed in-phase andanti-phase attractor.
1. Introduction
Recently, many researchers have shown their interests inchaotic systems [1][2][3]. In particular, chaos synchronization has attracted many researchers’ attentions and theirmechanism has been gradually made clear. This is because,chaotic circuit systems are good models to explain higher dimensional nonlinear phenomena in the field of natural science. Further, chaotic behavior has been expected in variousresearch fields. Hence, chaos has been investigated not only
in engineering but also in various fields such as medicine,sociology and economics. In the field of electrical and electronic engineering, researchers have investigated the phenomena in large scale coupling chaotic circuits. Moreover, they
proposed the new chaos-generating circuits and many 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 coupled chaos-generating circuits.
In our previous study, a simple chaotic oscillator using
two RC circuits was proposed [4]. When we change the parameter, we can observe not only periodic attractors but alsochaotic attractors in this simple oscillator. Further, we investigated the chaotic behavior when the number of the coupled
RC circuits are increased [5]. We investigated the attractors
and the Poincaré maps when we changed the value of parameter. In addition, we found the cross correlation characteristics between neighboring oscillators. In these previousstudies, we found that it is very interesting to investigate implementable chaotic circuit compose of RC circuit.

Figure 1: Circuit model

(a)

(b)

Figure 2: Rectangular voltage waveform

Figure 1 shows the circuit model. In this figure two chaoticoscillators, which was proposed in [4], are coupled via onecapacitor C0 . Two independent rectangular voltage sources
are connected to two comparators of each oscillator. Hence,
the whole circuit consists of two rectangular voltage sources,
four comparators, four resistors, four capacitors and one variable capacitor. Figure 2(a) shows the rectangular voltagewaveform VS (t). Eα is the amplitude of the rectangular voltage and T is the period of the waveform. E is the output volt-- 129 -age of the comparators, namely the DC supply voltage of theoperational amplifiers. The circuit equations are described asfollows:−v1 + E (v2 > VS 1 )dv1 
RC=dt −v − E (v < V )12(x − 1)e−τ + 1 (x4 > Vα ) 30
x3 =  (x + 1)e−τ − 1 (x < V )30
4α ( x20 + x40)( x20 − x40 )+ 1 e−τ −e−γτ − 122
(x3 > Vα )x4 = 
( x20 + x40)
( x20 − x40 )−τ
−1e−e−γτ + 1

22





(x3 < Vα )S1

C + C0C0−v2 −v4 − E


C+2C
C+2C0

0








(v1 > VS 1 )dv2 
RC=dt
C + C0C0−v2 −v4 + E
C+2C
C+2C0

0

(v1 < VS 1 )−v3 + E (v4 > VS 2 )dv3 
RC=dt −v − E (v < V )34S2
C + C0C0



−v2 −
v4 − E
C+2C
C+ 2C0

0





(v3 > VS 2 )

dv4 

RC=


dt

C0
C + C0
−v2 −v4 + E
C+2C
C+ 2C00

(v3 < VS 2 )where Vα corresponds to VS and is shown in Fig. 2(b). x10 ,x20 , x30 and x40 are initial values.
3. Simulation Results
We show the results of computer calculations. Moreover,Fig. 4 shows the phase diﬀerence between the left oscillator
and the right oscillator in Fig. 1.
3.1 Case of VS 1 = VS 2

Left attractors of Fig. 3 shows the left oscillator (x1 -x2 ) andcenter attractors shows the right oscillator (x3 -x4 ) in Fig. 1.
Right attractors of Fig. 3 shows the phase diﬀerence between
left oscillator and right oscillator (x1 -x3 ) in Fig. 1. In thiscomputer calculation, we use fixed parameters α = 0.060 and
β = 4.0. We observe the in-phase synchronization state in
By using the following variables and the parameters,
Fig. 3(1c) and when Fig. 3(5c), we observe the asynchronousstate. We find that when the value of the coupling strengthC0
vn = Exn , γ =, t = RCτ, T = RCβbecomes small, the strength of synchronization state of eachC + 2C0oscillator becomes weak in Fig. 3.
Next, we show the relationship between the coupling(n = 1, 2, 3, 4),strength and the degree of synchronization. We use fixed pawe obtain the normalized circuit equations. Because the cir- rameters α = 0.060 and β = 4.0. Figures 3(a) to (c) show
cuit equations are linear in each region, the rigorous solution the degree of synchronization by changing coupling strengthof the circuit equations can be derived as follows:
γ. Vertical axis is degree of synchronization D syn defined as−τfollows;(x − 1)e + 1 (x2 > Vα ) 10x1 = 
D syn = max{x1 − x3 } − min{x1 − x3 }.
 (x + 1)e−τ − 1 (x < V )102α ( x20 + x40)
( x20 − x40 )−τ
+1e+e−γτ − 1


22







(x1 > Vα )x2 = 
( x20 + x40)
( x20 − x40 )
− 1 e−τ +e−γτ + 1


22






(x1 < Vα )When D syn is small, our model is close to the synchronization
state. When the value of γ is small, we find that each oscillator do not become synchronization completely. However,the value of γ = 0.80, we observe the in-phase synchronization state. There is clearly a diﬀerence when one comparesγ = 0.05 and γ = 0.80.

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(1)(2)

Figure 4: Relation between coupling strength and degree ofsynchronization(3)

diﬀerence of the value of C0 in the case of VS 1 = −VS 2 . From
all of the time waveform, we find that chaos appears. FromFigs. 6(a) and (b), when the value of each rectangular voltage source is VS 1 = VS 2 , each oscillator becomes in-phasesynchronization. Compare Fig. 6(a) with (b), we observe inphase synchronization clearly in Fig. 6(a). By weakening thecoupling strength, we can observe the state, such as asynchronous. Figure 7 shows the attractor and time waveform
about the case of VS 1 = −VS 2 . We can not observe the antiphase synchronization clearly the case of VS 1 = −VS 2 . However, we can observe the attractor of anti-phase. Moreover,
we obtain the near results in Figs. 5(2c), (3c) and (5c).

(4)(5)(a)(b)(c)Figure 3: Attractor and phase diﬀerence in VS 1 = VS 2 (1) γ =0.80, (2) γ = 0.75, (3) γ = 0.50, (4) γ = 0.45 and (5) γ = 0.01
5. Conclusions
3.2 Case of VS 1 = −VS 2
Left attractors of Fig. 5 shows the left oscillator (x1 -x2 ) andcenter attractors shows the right oscillator (x3 -x4 ) in Fig. 1.
Right attractors of Fig. 5 shows the phase diﬀerence between
left oscillator and right oscillator (x1 -x3 ) in Fig. 1. In thiscomputer calculation, we use fixed parameters α = 0.060 and
β = 4.0. From Figs. 5(1c) and (4c), we observe the anti-phasesynchronization. Moreover, we observe the synchronization
and asynchronous state regardless of the value of couplingstrength.

4. Circuit Experiment

In this study, we have investigated two coupled chaotic oscillators composed of RC circuits. Two chaotic oscillators
were coupled by one capacitor C0 . We carried out computer calculation and circuit experiment about the case of
VS 1 = VS 2 and VS 1 = −VS 2 . In VS 1 = VS 2 of computer calculation, we confirmed that the relationship between couplingstrength and synchronization state. We found that when thevalue of the coupling strength becomes small, the strength ofsynchronization state of each oscillator becomes weak. Fromthe graph, we found the same relationship. In VS 1 = VS 2 , weobtained the diﬀerence phenomena from VS 1 = VS 2 . It regardless of the coupling strength, synchronization and asynchronous state are observed. In addition, we could be obtained the same result as computer calculation results by circuit experiment.

Figures 6 and 7 show the circuit experimental results.
Figs. 6(a) and (b) show the attractor and time waveform by
In our future work, we will make a detailed research of thisdiﬀerence of the value of C0 in the case of VS 1 = VS 2 . proposed circuit model. Moreover, we would like to investiFigs. 7(a) and (b) show the attractor and time waveform by gate lager scale circuits.

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(1)(a) C0 = 67.9[nF]

(2)

(3)
(b) C0 = 1.544[nF]
Figure 6: VS 1 = VS 2
(4)(5)(a) C0 = 48.6[nF]

(a)

(b)

(c)

Figure 5: Attractor and phase diﬀerence in VS 1 = −VS 2(1) γ = 0.80, (2) γ = 0.75, (3) γ = 0.50, (4) γ = 0.45 and(5) γ = 0.01(b) C0 = 10.0[nF]
Acknowledgment

Figure 7: VS 1 = −VS 2This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.
[4] S. Masuda, Y. Uchitani, and Y. Nishio: Simple Chaotic
Oscillator Using Two RC Circuits, Proc. of NCSP’09,
pp. 89-92, 2009.

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