2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Synchronization Phenomena of Two Chaotic Oscillators with Shifting Input Wave
Yoshinori Doike1 , Yoko Uwate1 , Yoshifumi Nishio1 and Jingmin Xin2
1Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone: +81-88-656-7470
E-mail: {doike,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.cn

Abstract

circuits with shifting input wave. Two chaotic oscillators arecoupled by one resistor or capacitor. We carry out computer
In this study, we investigate synchronization phenomena ob- calculation and investigate chaotic behavior when one side’sserved in two coupled chaotic oscillators with shifting input input wave is shifted.
wave. In this circuit system, two chaotic circuits are coupledby one resistor or capacitor. We investigate the synchronization state when one side’s input wave is shifted. We carry out
2. Circuit Modelcomputer calculation to investigate in-phase and anti-phaseusing correlation value. Moreover, when we change the value
Figures 1, 2 show the circuit model in this study. Twoof coupling strength between two oscillators, we observe the chaotic oscillators, which was proposed in [4], are coupledchange of synchronization state.
via one resistor or capacitor. An independent rectangularvoltage sources VS1 and VS2 are connected of each oscillator’s two comparators.

1. Introduction
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. Chaotic behavior has
been observed in various fields of research. Hence, chaos has
been investigated not only in engineering but also in various
fields such as biology, sociology and economics. In the fieldof electrical and electronic engineering, many researchers has
proposed many applications using chaos such as chaos communications, chaos cryptosystem and chaos neural networks.
In order to realize chaotic engineering systems, it is importantto understand simple coupled chaos-generating circuits.
In our previous study, a simple chaotic oscillator composedof two RC circuits were proposed [4]. When we changed theparameter, we could observe not only periodic attractor but
also chaotic attractor in this simple oscillator. And, we investigated synchronization phenomena when two chaotic oscillator composed of RC circuits are coupled by one resister.
Moreover, we found the cross correlation characteristics between neighboring oscillators. In addition, we have investigated synchronization phenomena when two input waves arein-phase and anti-phase [5].
In this study, we investigate synchronization phenomena
observed in two coupled chaotic oscillators composed of RCv2

RR

Cv1

Cv

4v3

vS1vS2
R

R

r

v3

v1
Cv

C2

v

4

Figure 1: Circuit model 1.

v2

RRCv1

Cv

4v3vS1vS2
c0R

Rv3

v1
Cv

2Cv

4

Figure 2: Circuit model 2.
Hence, the whole circuit consists of two rectangular voltage sources, four comparators, four resistors, one variable resistor and four capacitors. Figure 3(a) shows the rectangularvoltage waveform VSn (t) (n = 1, 2). Eα is the amplitude of
the rectangular voltage and T is the period of the waveform.
E is the output voltage of the comparators, namely the DC
supply voltage of the operational amplifiers.

- 185 -VSn(t)Vαn(τ)Tβ) (x + x2040
+ 1 e−τ2(x − x )2040+e−γτ − 1 (x1 > Vα1 )αEατ

tx2 =－α－Eα(a)(b)

Figure 3: Rectangular voltage waveform.
Figure 1 is circuit model which composed two chaotic oscillators coupled by one resistor. Normalized system equations are given as follows: (x10 − 1)e−τ + 1 (x2 > Vα1 )
x1 =

−τ

x3 =)
 (x + x2040
+ 1 e−τ


2



(x20 − x40 ) −(1+2γ)τ

e− 1 (x1 > Vα1 ) +2x4 =−1(x10 + 1)ex2 =(x2 < Vα1 )(x4 > Vα2 )(x4 < Vα2 )(x30 + 1)e−τ − 1) (x + x2040
+ 1 e−δτ2(x20 − x40 ) −(1+2γ)δτ
e− 1 (x3 > Vα2 ) +x4 =2)(x + x2040

−1e−δτ
2




()

 + x20 − x40 e−(1+2γ)δτ + 1 (x < V )3α220 + x402+)− 1 e−τ(x20 − x402)e−γτ + 1 (x1 < Vα1 ) (x30 − 1)e−τ + 1(x4 > Vα2 )(x4 < Vα2 )(x30 + 1)e−τ − 1) (x + x2040
+ 1 e−δτ2(x − x )2040

+e−γδτ − 1 (x3 > Vα2 )2(x + x)
2040

− 1 e−δτ


2




(x − x )2040

+e−γδτ + 1 (x3 < Vα2 )(2)where Vαn is a parameter corresponding to VSn (n = 1, 2), β
is parameter corresponding to T and γ is a coupling strength.
x10 , x20 , x30 and x40 are initial values.
Because these circuit equations are linear in each region, therigorous solution of these circuit equations can be derived.
We carry out computer calculation, the value of the frequency error δ is fixed with 0.05% to remove synchronization
isn’t due to chaotic phenomena.

2

x3 =













2(x + x)


2040

−1e−τ


2





()

 + x20 − x40 e−(1+2γ)τ + 1 (x < V )1α1 (x30 − 1)e−τ + 12

 (x3. Simulations(1)

2

where Vαn is a parameter corresponding to VSn (n = 1, 2), β
is parameter corresponding to T and γ is a coupling strength.
x10 , x20 , x30 and x40 are initial values.

First, we explain 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. 4 shows two input waves when gap is 2.50.
Second, we explain how to judge in-phase synchronization and anti-phase synchronization. To judge in-phase synchronization and anti-phase synchronization quantitatively,
we use correlation coefficient in this study. Correlation coefficient is described as follows:1∑(x1 (i) − x1 )(x2 (i) − x2 )k i=0vc= v
u ku ku1 ∑u1 ∑2t(x1 (i) − x2 ) t(x2 (i) − x2 )2k i=0
k i=0kFigure 2 is circuit model which composed two chaotic oscillators coupled by one capacitor. Normalized system equations are given as follows: (x10 − 1)e−τ + 1 (x2 > Vα1 )x1 =−τ(x10 + 1)e−1(x2 < Vα1 )(3)where x1 and x2 are state variables and k is number of trials.
If c is near 1.0, synchronization state is in-phase. In contrast,
if c is near −1.0, synchronization state is anti-phase.

- 186 -βV1gap = 0.00
gap

V2
gap = 1.25

tFigure 4: Two input wave (gap = 2.50).
gap = 2.50

4. Simulation Results
We show the results of the computer calculations and use
fixed parameters α = 0.06 and β = 10.
First, we show the result of circuit model 1 when the coupling strength are 0.1 and 1.0 and circuit model 2 when the
coupling strength is 0.5.

gap = 0.00

gap = 3.75

gap = 5.00
(a)gap = 1.25

(b)

(c)

Figure 6: Chaotic attractor of circuit model 1 obtained fromcomputer calculation (γ = 1.0). (a) x1 − x2 , (b) x3 −x4 , and (c) x1 − x3 .

attractors show the right oscillator (x3 − x4 ). Right attractors
show the phase difference between left and right oscillators(x1 − x3 ).
gap = 2.50
We observe the in-phase synchronization state in Figs. 6(c)and 7(c). These results are caused by two rectangular voltages are in-phase relations. Furthermore, we can observe theanti-phase synchronization state in Fig. 5(c) and 7(c). Moreover, we observe in-phase and anti-phase synchronization in
gap = 3.75
same parameter when two oscillators are coupled by capacitor. However, we can not observe in-phase and anti-phasesynchronization in same parameter when two oscillators arecoupled by resister.
Next, Fig. 8 shows the relation between ”gap” and correlagap = 5.00
tion coefficient between the two oscillators in Figs. 1 and 2.
In this computer simulation, we define k = 100000. And, we(a)(b)(c)
calculate average of solution to derive correlation values.
We obtain correlation values is small when ”gap” is large
Figure 5: Chaotic attractor of circuit model 1 obtained fromcomputer calculation (γ = 0.1). (a) x1 − x2 , (b) x3 − from Fig. 8. This result is caused by two rectangular voltages are gradually anti-phase relations. Moreover, it becomes
x4 , and (c) x1 − x3 .
synchronization state when the value of coupling strength is
Figures 5 to 7 shows attractor and the phase difference. large. And, we obtain when the value of coupling strength
Left attractors show the left oscillator (x1 − x2 ) and center is small, we observe anti-phase synchronization. There is- 187 -gap = 0.00

gap = 1.25

gap = 2.50(a) Circuit model 1.

gap = 3.75

gap = 5.00
(a)

(b)

(c)

Figure 7: Chaotic attractor of circuit model 2 obtained fromcomputer calculation (γ = 0.5). (a) x1 − x2 , (b) x3 −x4 , and (c) x1 − x3 .

(b) Circuit model 2.

Figure 8: Relation between ”gap” and correlation values.
clearly a difference when one compares γ = 0.1 and γ = 1.0 observed in-phase and anti-phase synchronization in the samein Fig. 8(a). In Fig 8(a), when γ is large, we observe in- parameter. When two chaotic oscillators were coupled by caphase synchronization. However, we can not observe anti- pacitor, we observed in-phase and anti-phase synchronization
phase synchronization when γ is large. By contrast, when in the same parameter.
In our future works, we investigate to change input squareγ is small, we observe anti-phase synchronization. However,
we can not observe in-phase synchronization when γ is small. wave into other wave as sin wave and saw wave.
From these results, we find it is impossible to observe changReferences
ing into in-phase synchronization to anti-phase synchronization or into anti-phase synchronization to in-phase synchro- [1] T. Inagaki and T. Saito, “Consistency in a Chaotic Spiking
Oscillator,” IEICE Trans. Fundamentals, vol. E91-A, no. 8,
nization by shifting input wave when two chaotic oscillators
pp. 2240-2243, 2008.
are coupled resistor. Furthermore, all locus are similar.
Figure 8(b) is almost similar result to Fig. 8(a) without [2] B. Mutheswamy, “Implementing Memristor Based Chaotic Circuits,” International Journal of Bifurcation and Chaos, vol. 20,
γ = 0.5. We observe in-phase and anti-phase synchronizano. 5, pp. 1335-1350, 2010.
tion when γ = 0.5.
5. Conclusion
In this study, we have investigated synchronization phenomena observed in two coupled chaotic oscillators composed with shifting input wave. Two chaotic oscillators werecoupled by resistor or capacitor. We carried out computer calculation to investigate. We could observe in-phase and antiphase synchronization by using correlation coefficient. Whentwo chaotic oscillators were coupled by resistor, we could not[3] B. Mutheswamy and L.O. Chua, “Simplest Chaotic Circuit,”
International Journal of Bifurcation and Chaos, vol. 20, no. 5,pp. 1567-1580, 2010.
[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, Y. Uwate, Y. Nishio and J. Xin, “Synchronization
Phenomena of Two Simple RC Chaotic Circuits Composed bya Capacitor,” Proc. of NCSP’13, pp. 129-132, 2013.

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