IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2013

Synchronization Phenomena of Two Chaotic
Circuits with Shifting Input Wave
Yoshinori Doike† , Yoko Uwate† , Yoshifumi Nishio† and Jingmin Xin‡
† Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {doike, uwate, nishio}@ee.tokushima-u.ac.jp‡ Dept. of Electrical and Electronic Engineering, Xi’an Jiaotong University
28 Xianning Xilu, Xi’an, Shaanxi 710049, China
Email: jxin@mail.xjtu.edu.cn

Abstract—In this study, we investigate synchronization phenomena observed in two coupled chaotic oscillators with shifting
the input wave. In this circuit system, two chaotic circuits arecoupled by one resistor. We investigate the synchronization state
when one side’s input wave is shifted. We carry out computer
calculation to investigate in-phase and anti-phase.

2. C IRCUIT M ODEL
Figure 1 shows the circuit model in this study. Two chaotic
oscillators, which was proposed in [4], are coupled via one
resistor. An independent rectangular voltage sources VS1 andVS2 are connected of each oscillator’s two comparators.

v2

1. I NTRODUCTION
Recently, many researchers have shown their interests inchaotic systems [1]-[3]. In particular, chaos synchronization
has attracted many researchers’ attentions and their mechanism has been gradually made clear. This is because, chaotic
systems observing chaos synchronization are good models
to explain and describe the higher dimensional nonlinear
phenomena in the field of natural science. Further, chaotic
behavior has been expected in various fields of research.
Hence, chaos has been investigated not only in engineering
but also in various fields such as medicine, biology sociology
and economics. In the field of electrical and electronic engineering, many applications using chaos has been proposed bymany researchers. For example, chaos communications, chaos
cryptosystem and chaos neural networks. In order to realize
chaotic engineering systems, it is important to understandsimple coupled chaos-generating circuits.
In our previous study, a simple chaotic oscillator composedof two RC circuits were proposed [4]. When we changed
the parameter, we could observe not only periodic attractor
but also chaotic attractor in this simple oscillator. Further,
we investigated the chaotic behavior when the number of thecoupled RC circuits are changed [5]. Moreover, we foundthe cross correlation characteristics between neighboring oscillators. In addition, we have investigated synchronization
phenomena when two input waves are in-phase and anti-phase[7].
In this study, we investigate synchronization phenomena
observed in two coupled chaotic oscillators composed of RC
circuits with shifting input wave. Two chaotic oscillators arecoupled by one resistor. We carry out computer calculationand investigate chaotic behavior when one side’s input waveis shifted.

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RRCv1 v3CvS2vS1Rv1CFig. 1.

VSn(t)

Rrv

2vv3
C

4Circuit model.

Vαn(τ)TβαEα

τt－α－Eα(a)Fig. 2.

(b)
Rectangular voltage waveform.

Hence, the whole circuit consists of two rectangular voltage
sources, four comparators, four resistors, one variable resistor
and four capacitors. Figure 2(a) shows the rectangular voltage
waveform VSn (t) (n = 1, 2). Eα is the amplitude of therectangular voltage and T is the period of the waveform. E isthe output voltage of the comparators, namely the DC supply
voltage of the operational amplifiers.

where Vαn is a parameter corresponding to VSn (n = 1, 2),
β is parameter corresponding to T and γ is a couplingstrength. x10 , x20 , x30 and x40 are initial values. In thiscomputer calculation, we add the frequency error, in order
to set up same condition to the circuit experiment. We carryout computer calculation, the value of the frequency error δis fixed with 0.05% to remove synchronization isn’t due tochaotic phenomena.

The circuit equations are described as follows:(v2 > VS1 ) −v1 + E
dv1RC=
dt−v1 − E(v2 < VS1 )R −v2 + r (v4 − v2 ) − E (v1 > VS1 )dv2RC=dt −v + R (v − v ) + E (v < V )2421S2r
(v4 > VS2 ) −v3 + E
dv3RC=
dt−v3 − E(v4 < VS2 )R(v3 > VS2 ) −v4 + r (v2 − v4 ) − Edv4RC=dt −v + R (v − v ) + E(v3 < VS2 )424r

3. S IMULATIONS
We explain the input wave. The input wave’s period β is 10.
The step size of τ is 0.01. We define deviation of two input
waves as ”gap”. In this study, we investigate synchronization
state when the gap is changed from 0 to 5. For example, Fig. 3
shows two input waves when gap is 2.5.

(1)β

by using the following variables and the parameters,

V1R
vi = Exi , t = RCτ, T = RCβ,= γ,r(i = 1, 2, 3, 4)

gap

we obtain the normalized circuit equations. Because the circuit
equations are linear in each region, the rigorous solution of thecircuit equations can be derived as follows:x1 = (x10 − 1)e−τ + 1(x2 > Vα1 )

(x2 < Vα1 )(x10 + 1)e−τ − 1V2Fig. 3.

)
 (x + x2040
+1e−τ2(x − x )2040
+e−(1+2γ)τ − 1 (x1 > Vα1 )
2x2 =)
 ( x20 + x40
− 1 e−τ2(x − x )2040

+e−(1+2γ)τ + 1 (x1 < Vα1 )2{(x30 − 1)e−τ + 1(x4 > Vα2 )x3 =(x30 + 1)e−τ − 1(x4 < Vα2 )) (x + x2040
+1e−δτ

2




(x − x )2040

e−(1+2γ)δτ − 1+
2x4 =)
 ( x20 + x40−1e−δτ

2



()

 + x20 − x40 e−(1+2γ)δτ + 12t

Two input wave (gap = 2.5).

4. S IMULATION R ESULTS
We show the results of the computer calculations. We use
fixed parameters α = 0.061 and β = 10.
Figure 4 show attractor and the phase difference. Left
attractors show the left oscillator (x1 −x2 ) and center attractors
show the right oscillator (x3 − x4 ) in Fig. 1. Right attractors
show the phase difference between left oscillator and rightoscillator (x1 − x3 ) in Fig. 1. We can observe the anti-phasesynchronization state in Fig. 4(c). This result is caused by
two rectangular voltages are anti-phase relations. From theseresults, we find that it is impossible to observe changing
into anti-phase synchronization to in-phase synchronization byshifting input wave when two chaotic oscillators composed RCcircuit are coupled via resistor.

(x3 > Vα2 )(2)(x3 < Vα2 )- 15 -

[4] S. Masuda, Y. Uchitani, and Y. Nishio, “Simple Chaotic Oscillator Using
Two RC Circuits,” Proc. of NCSP’09, pp. 89-92, 2009.
[5] A. Shimada, H. Kumeno, Y. Uwate, Y. Nishio, and J. Xin, “CoupledChaotic Oscillators Composed of RC Circuits,” Proc. of NCSP’12, pp. 8891, 2012.
[6] A. Shimada, Y. Uwate, Y. Nishio, and J. Xin, “Analysis of Fifteen Coupled
Chaotic Oscillators Composed of RC Circuits,” Proc. of NOLTA’12,
pp. 785-788, 2012.
[7] A. Shimada, Y. Uwate, Y. Nishio, and J. Xin, “Synchronization Phenomena of Two Simple RC Chaotic Circuits Composed by a Capacitor,” Proc.
of NCSP’13, pp. 129-132, 2013.

gap = 0gap = 1.25

gap = 2.5

gap = 3.75

gap = 5.0(a)(b)(c)

Fig. 4. Chaotic attractor obtained from computer calculation (γ = 0.1).
(a) x1 − x2 , (b) x3 − x4 , and (c) x1 − x3 .

5. C ONCLUSION
In this study, we have investigated synchronization phenomena observed in two coupled chaotic oscillators composed of
RC circuits with shifting input wave. Two chaotic oscillators
were coupled by one resistor. We carried out computer calculation to investigate.
We confirmed that if the value of coupling strength is small,anti-phase synchronization can be observed. Furthermore, inphase synchronization can not be observed.
In our future works, we investigate other circuit modelcomposed two coupled chaotic oscillators be coupled by one
capacitors. Then, we would like to experiment with electric
circuit at the first time. Next, we will investigate larger scale
circuits, because this model is very simple structure.
R EFERENCES[1] T. Inagaki and T. Saito, “Consistency in a Chaotic Spiking Oscillator,”
IEICE Trans. Fundamentals, vol. E91-A, no. 8, pp. 2240-2243, 2008.
[2] B. Mutheswamy, “Implementing Memristor Based Chaotic Circuits,”
International Journal of Bifurcation and Chaos, vol. 20, no. 5, pp. 13351350, 2010.
[3] B. Mutheswamy and L,O. Chua, “Simplest Chaotic Circuit,” International
Journal of Bifurcation and Chaos, vol. 20, no. 5, pp. 1567-1580, 2010.

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