Journal of Signal Processing, Vol.17, No.4, pp.99-102, July 2013

SELECTED PAPER

Synchronization Phenomena of Two Simple RC Chaotic Circuits Coupled by a Capacitor
Ayana Shimada1 , Yoko Uwate1 , Yoshifumi Nishio1 and Jingmin Xin2
1

Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Phone: +81-88-656-7470, FAX: +81-88-656-7471
E-mail: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
In this study, two coupled chaotic oscillators composed of RCcircuits are investigated. We carry out computer simulations
and circuit experiments and observe both in-phase synchronization and anti-phase synchronization. Moreover, when wechange the coupling strength between oscillators, we observea change in synchronization state.
1. Introduction
Recently, many researchers have shown interest in chaotic
systems [1],[2]. In particular, chaos synchronization has attracted many researchers’ attention and its mechanism has
been gradually clarified [3],[4]. In the field of electrical andelectronic engineering, researchers have proposed various applications using chaos, for example, chaos communication
systems [5], chaos cryptosystems [6], and so forth. In orderto realize such chaotic engineering systems, it is important toinvestigate simple coupled chaos-generating circuits.
In our previous study, a simple chaotic oscillator using
two RC circuits was proposed [7]. When we change the parameter, we can observe not only periodic attractors but alsochaotic attractors from this simple oscillator. Furthermore,
we investigated the chaotic behavior by increasing when thenumber of coupled RC circuits [8]. In addition, we focusedon the cross-correlation characteristics between neighboring
oscillators. Because this chaotic oscillator has an extremely
simple structure, it is important to investigate its basic nonlinear phenomena.
In this study, we investigate synchronization phenomena
observed from two chaotic oscillators composed of RC circuits coupled by a capacitor. We carry out computer simulations and circuit experiments for the two cases in which theexternal forces have the same phase states and inverse phasestates.

2Xi’an Jiaotong University
28 Xianning Xilu, Xi’an, Shaanxi 710049, China
Phone: +86-29-8266-8802
E-mail: jxin@mail.xjtu.edu.cn

C0 . Two independent rectangular voltage sources are connected to the two comparators of the oscillators. Hence, thewhole circuit consists of two rectangular voltage sources, fourcomparators, four resistors, four capacitors, and one coupling
capacitor. Figure 2(a) shows the rectangular voltage signals
VS (t). Eα is the amplitude of the rectangular voltage and T
is the period of the waveform. E is the output voltage ofthe comparators, namely, the DC supply voltage of the operational amplifiers. The circuit equations are described asfollows: −v1 + E
dv1 

RC=

 −v1 − Edt(v2 > VS 1 )(v2 < VS 1 ) C +CC00v2 −v4 − E−
C + 2C0
C + 2C0
(v1 > VS 1 )dv2 
=
RC
C+CCdt

00


−v2 −
v4 + E


C+2C
C+2C

00

(v1 < VS 1 )



dv3 
 −v3 + E=
RC −v3 − Edt(v4 > VS 2 )(v4 < VS 2 )
C + C0C0v2 −v4 − E−
C+2C
C+ 2C00(v3 > VS 2 )dv4 
=RC

C0C + C0dt


−v2 −
v4 + E
C+2C
C+ 2C00(v3 < VS 2 )

(1)

(2)

(3)

(4)

By using the following variables and the parameters
2. Circuit Model
Figure 1 shows the circuit model. In this figure, two chaotic
oscillators, as proposed in [6], are coupled via one capacitor

Journal of Signal Processing, Vol. 17, No. 4, July 2013

vn = Exn , γ =C0, t = RCτ, T = RCβ
C + 2C0(5)(n = 1, 2, 3, 4)99

Figure 1: Circuit model
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:−τ(x2 > Vα ) (x10 − 1) e + 1
x1 = (6) (x10 + 1) e−τ − 1(x2 < Vα )(x − x ) (x + x)
2040
2040

−τ
+1ee−γτ − 1+22

(x1 > Vα )x2 = 
(()
x20 − x40 ) −γτ
x20 + x40−τ−1ee +1+
22


(x1 < Vα )−τ (x30 − 1) e + 1x3 =  (x30 + 1) e−τ − 1(x4 > Vα )(x4 < Vα ))(x − x ) (x + x20402040
−τ
+1ee−γτ − 1−
22



(x3 > Vα )x4 = ))((x20 − x40 −γτ
x20 + x40
− 1 e−τ −e +122(x < V )
3(a)
(7)

(8)

(b)
(9)

α

where Vα corresponds to VS and is shown in Fig. 2(b). x10 ,x20 , x30 , and x40 are the initial values in the corresponding
linear regions.
3. Computer SimulationsFirst, we present and discuss the computer-simulated results. In the following results, the parameters corresponding to the rectangular voltage signals are fixed as α = 0.060
and β = 4.0 and the parameter corresponding to the couplingstrength γ is varied in order to investigate the synchronization
phenomena of chaotic oscillations.

100

Figure 2: Rectangular voltage signals

3.1 Case of VS 1 = VS 2
Figure 3 shows computer-simulated results for the casein which the two rectangular voltage signals are the same,
namely VS 1 = VS 2 . Figures 3(a) and 3(b) show the attractor(x1 − x2 ) observed from the left oscillator in Fig. 1 and theattractor (x3 − x4 ) observed from the right oscillator in Fig. 1,
respectively. Figure 3(c) shows the phase states between thetwo attractors (x1 − x3 ).
For strong-coupling parameters, we can observe complete
in-phase synchronization as shown in Fig. 3(1), although thetwo oscillators exhibit chaotic oscillations. As decreasing
the coupling parameter γ, the in-phase synchronization state

Journal of Signal Processing, Vol. 17, No. 4, July 2013

(1)

(2)

(3)

(4)

Figure 4: Relationship between coupling strength and degreeof synchronization−VS 2 . It is interesting to investigate how the two coupledoscillators behave for the inverse phase external signals, because such a complicated situation was not considered in paststudies. Figure 5 shows the computer-simulated results. Figures 5(a) and 5(b) show the attractors (x1 − x2 ) and (x3 − x4 ),respectively. Figure 5(c) shows the phase states between thetwo attractors (x1 − x3 ).
For strong-coupling parameters, we can observe complete
anti-phase synchronization as shown in Fig. 5(1). As decreasing the coupling parameter γ, the anti-phase synchronization
state becomes weak, similar to the case of VS 1 = VS 2 .
4. Circuit Experiments(5)

Finally, we show the circuit experimental results. Figures 6(a) and 6(b) show the phase states and time waveforms
for the case of VS 1 = VS 2 and Figs. 7(a) and 7(b) show thephasestates and time waveforms for the case of VS 1 = −VS 2 .
(a)
(b)
(c)
From all of the time waveforms, we can confirm thatFigure 3: Attractors and phase diﬀerence for VS 1 = VS 2 : chaotic oscillations appear in the circuit. We also observe in(1) γ = 0.80, (2) γ = 0.75, (3) γ = 0.50, (4) γ = 0.45, phase synchronization and anti-phase synchronization similar
to those in the computer-simulated results.
and (5) γ = 0.01
5. Conclusions
gradually becomes weak and changes to an asynchronous
In this study, we have investigated two coupled chaoticstate.
oscillatorscomposed of RC circuits. We carried out comIn order to evaluate the synchronization states quantitaputer
simulations
and circuit experiments for the two cases oftively, we introduce the degree of synchronization by usingV=VandVS
1S
2S
1 = −VS 2 . For the case of VS 1 = VS 2 , wethe distance between the two solutions as follows:
confirmed the generation of in-phase synchronization states,
D syn = max{|x1 − x3 |} − min{|x1 − x3 |}(10) while for the case of VS 1 = −VS 2 , we confirmed the generation of anti-phase synchronization states.
Figure 4 shows the relationship between the coupling strength
Detailed research on the synchronization states such as staand the degree of synchronization. We can see that the degree
bility analysis and the coexistence region of the in-phase andof synchronization immediately changes after breaking theanti-phase states will be carried out as future research.
synchronization state.
3.2 Case of VS 1 = −VS 2

Next, we consider the case in which the two rectangular voltage signals have inverse phase states, namely VS 1 =Journal of Signal Processing, Vol. 17, No. 4, July 2013

Acknowledgments
This work was partly supported by a JSPS Grant-in-Aid for
Scientific Research (No.22500203).

101

(1)

(a) C0 = 67.9[nF](2)(3)(b) C0 = 1.5[nF]
Figure 6: Circuit experimental results for VS 1 = VS 2(4)(5)(c) C0 = 48.6[nF]
(a)

(b)

(c)

Figure 5: Attractors and phase diﬀerence for 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(d) C0 = 10.0[nF]
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Journal of Signal Processing, Vol. 17, No. 4, July 2013