2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Partial Decline of Synchronization Ratein Chaotic Circuits Coupled by a Resistor
Tatsuki Sobue† , Yasuteru Hosokawa‡ , Yoko Uwate† and Yoshifumi Nishio††Dept. Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email:{sobue, uwate, nishio}@ee.tokushima-u.ac.jp,‡Dept. Media and Information Syst., Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp
Abstract—In our previous study, a modified ShinrikiMori circuit was proposed. The circuit includes two LCresonators which are coupled by resistors. By changing
a parameter, switching phenomena of synchronization andasynchronization between two resonators were observed.
In this study, we propose new coupled system which two
previous study’s systems are coupled by a resistor. As aresult, interesting phenomena about relationship between
LC resonators and coupling strength are observed.

Circuit-AGai
G A1f( gA1 , vA3 - vA1) v A1

A1

C A1

L A1f( gA2 , vA3 - vA2)

v

A3i
G A2v

A2A2C A2L A2Gi1. Introduction
In coupled chaotic circuits, many interesting phenomena, for instance, chaotic synchronization phenomena, intermittency chaos, spatio-temporal chaos and so on are observed. Therefore, there are many studies about coupledchaotic circuits. In almost these studies, some of famous
chaotic circuits have been applied. One of famous chaoticcircuits is a Shinriki-Mori circuit [1] [2]. There are many
investigations of coupled chaotic circuits using a ShinrikiMori circuit [4] [5].
In our previous study, a modified Shinriki-Mori circuit[6] has been proposed. The Shinriki-Mori circuit consistsof a LC resonator, by-directional diodes and the others.
In the proposed circuit, a LC resonator and by-directional
diodes included in the original circuit are copied and added
to the original circuit. By applying diﬀerent parameters ofLC resonators, two similar chaotic waveforms which areobserved as voltages of LC resonators are obtained. Additionally, these waveforms are influenced by changing circuit parameters.
In this study, two modified Shinriki-Mori circuits coupled by a resistor are proposed. This system includes parameter mismatches. Therefore, LC resonators could notsynchronized at all. In this system, similar chaotic waveforms are also observed and these are not synchronizedat all. Additionally, these waveforms are influenced bychanging circuit parameters. Namely, it means that inspite of asynchronous state, the relationship among LC resonators exist. The aim of this study is to reveal the relationship.

G B1f( gB1 , vB3 - vB1) v B1
GbB1

C B1

L B1f( gB2 , vB3 - vB2)v

B3i
G B2Circuit-Bv

B2

B2

C B2

L B2

Figure 1: Proposed System.

2. Proposed System
Figure 1 shows the proposed system. Two modifiedShinriki-Mori circuits are coupled by a resistor. In thisstudy, two modified Shinriki-Mori circuits are called as Circuit A and Circuit B, respectively. Additionally, LC resonators included in two circuits are called as subcircuit A1,A2, B1 and B2, respectively. A diﬀerence between Circuit
A and Circuit B is only parameters of subcircuits. In orderto change parameters of nonlinearity, resistors G A1 , G A2 ,G B1 and G B2 are added. Bidirectionally coupled diodes aremodeled as a piecewise linear function shown in Figure 2.
The others are modeled as linear elements.
Using this model, the circuit equation is described as follows.

- 205 -v

By substituting the variables and the parameters,√v{a1,a2,a3}
La1 i{a1,a2}, x{A4,A5} =,x{A1,A2,A3} =Vth
Ca1 Vth√v{b1,b2,b3}
La1 i{b1,b2}
x{B1,B2,B3} =, x{B4,B5} =,Vth
Ca1 Vth√dLa1= “ · ”, α{a1,a2,a3,b1,b2,b3} = g{a1,a2,a3,b1,b2,b3},dt
Ca1Ca1La1
β{a2,a3,b1,b2,b3} =, γ{a2,b1,b2} =,C{a2,a3,b1,b2,b3}
L{a2,b1,b2}√√
La1La1
δ{a,b} = G{a,b}, ε=G,Ca1
Ca1i(a)

(b)iv

(c)

(3)
equations (1) and (2) are described as follows.

Figure 2: Bidirectionally coupled diodes model. (a) Circuit
schematic. (b) Circuit model. (c) v − i characteristic.

dva1


Ca1dt

dv

a2

Ca2

dt



dva3



Ca3



dt




dia1



La1



dt




di
a2


La2



dt




dvb1



Cb1



dt






dvb2


Cb2



dt





dvb3


Cb3


dt







dib1


Lb1


dt

dib2L b2 dt=ga1 f (va3 − va1 ) − ia1 ,=ga2 f (va3 − va2 ) − ia2 ,= Ga va3 − ga1 f (va3 − va1 )−ga2 f (va3 − va2 ) − G(va3 − vb3 ),=va1 ,=va2 ,=gb1 f (vb3 − vb1 ) − ib1 ,=gb2 f (vb3 − vb2 ) − ib2 ,(1)=vb2 ,

wheref (v) = v + (|v − vth | − |v + vth |)/2.

αa1 f ′ (xA3 − xA1 ) − xA4 ,{}βa2 αa2 f ′ (xA3 − xA2 ) − xA5 ,{βa3 δa xA3 − αa1 f ′ (xA3 − xA1 )

=
=
=
=
=

−αa2 f ′ (xA3 − xA2 ) − ε(xA3 − xB3 ) } ,xA1 ,γa2 xA2 ,{}βb1 αb1 f ′ (xB3 − xB1 ) − xB4 ,{}βa2 αb2 f ′ (xB3 − xB2 ) − xB5 ,{βb3 δb xB3 − αb1 f ′ (xB3 − xB1 )==−αb2 f ′ (xB3 − xB1 ) + ε(xA3 − xB3 ) } ,γb1 xB1 ,γb2 xB2 ,(4)f ′ (x) = x + (|x − 1| − |x + 1|)/2.

(5)

3. Computer Simulations−gb2 f (vb3 − vb2 ) + G(va3 − vb3 ),
vb1 ,

=
=
=

where= Gb vb3 − gb1 f (vb3 − vb1 )

=


ẋA1

ẋA2
ẋA3




ẋA4





 ẋA5ẋB1
ẋB2
ẋB3

ẋB4
 ẋB5(2)In our previous study, a modified Shinriki-Mori circuit
was investigated [6]. Following results were obtained. In
the case of applying same parameters to subcircuits, subcircuits are synchronized. Additionally, switching phenomena between synchronization and asynchronization are observed in the case of applying diﬀerent circuit parametersof subcircuits. As a result of investigations of the relationship between synchronization rates and circuit parameters, it was confirmed that some similar chaotic waveforms
could be obtained at the same time.
In this study, two modified Shinriki-Mori circuits coupled by a resistor are investigated. This system includes
four subcircuits. Additionally, four subcircuits are notequivalent relations. Therefore, the relationship amongfour subcircuits is very interesting.
Figure 3 shows one of computer simulation results.
Double scroll type attractors can be observed. Double- 206 -xA1 - xA2
xA2 - xB1
xB1 - xB2(1) ε=0.1(1) xA3 - xA1(2) xB3 - xB1

Figure 3: Attractors of Circuit A and Circuit B. αa1 = αa2= αb1 = αb2 = 9, βa2 = 0.8, βb1 = 0.4, βb2 = 0.7, βa3 = βb3 =0.32, γa2 = γb1 = γb2 = 1.0 and δa = δb = 1.4. (1) Circuit A.
Horizontal axis is xA3 and vertical axis is xA1 . (2) Circuit
B. Horizontal axis is xB3 and vertical axis is xB1 .

xA1 - xA2xA2 - xB1xB1 - xB2(2) ε=0.7xA1xA2xA1 - xA2xA1 - xA2xA2 - xB1xB1 - xB2

Figure 4: Waveforms of subcircuit A1 and A2. αa1 = αa2 =αb1 = αb2 = 9, βa2 = 1.0, βb1 = 0.4, βb2 = 0.7, βa3 = βb3 =0.32, γa2 = γb1 = γb2 = 1.0 and δa = δb = 1.4.
scroll type attractor of Figure 3(2) is larger attractor of Figure 3(1). In this case, waveforms of subcircuit A1 and A2
are shown in Figure 4. Quasi-synchronization and asynchronization states are appear at randomly. In order to investigate this phenomenon, the relationship between voltage diﬀerences of subcircuits and a coupling strength ispaid attention. Figure 5 shows the relationship between
the voltage diﬀerences and the coupling strength. The voltage diﬀerence between vA1 - vA2 is corresponding to xA1 xA2 . In the same way, subcircuit A2 and B1, B1 and B2
are corresponding to xA2 - xB1 and xB1 - xB2 , respectively.
In among these subcircuits, synchronization and asynchronization states are observed. By increasing the couplingstrength ε, synchronization rate of xA1 - xA2 is decreased inFigure 5(1)-(3). On the other hand, synchronization rate ofxB1 - xB2 is increased.
In order to make this phenomenon clear, a definition of
the synchronization is defined as follows.
|xA1 − xA2 | < 0.1|xA2 − xB1 | < 0.1|xB1 − xB2 | < 0.1(6)Equation 6 shows a definition of the synchronization asshown in Figure 5. Namely, a range of the synchronous
state is from 0.1 to -0.1. Using this definition, the relation-(3) ε=1.0
Figure 5: Diﬀerential waveforms of Circuit A and Circuit
B. αa1 = αa2 = αb1 = αb2 = 9, βa2 = 0.8, βb1 = 0.4, βb2 =0.7, βa3 = βb3 = 0.32, γa2 = γb1 = γb2 = 1.0 and δa = δb =1.4.

ship between synchronous states and coupling strength isinvestigated.
Figure 7 shows the statistical data of synchronous rates
and the coupling strength ε. A step size of the couplingstrength ε is 0.01. Calculation time of each parameter is500. Time step is 0.005. The synchronous rate between
subcircuits A1 and A2 is decreasing by increasing couplingstrength ε from 0 to 1. After that the rate is increasing andbecomes 100%. On the other hand, the synchronous ratebetween subcircuits B1 and B2 is increasing by increasing coupling strength ε. Additionally, the synchronous rate
between subcircuits A2 and B1 remains very low rate. Inε > 1.3, the rate becomes 100%. In 0.0 < ε < 1.0, double scroll type attractors can be observed in two circuits.
However, In ε > 1.0, oscillations becomes periodic orbits.
Therefore, rates become 100%.
This result shows a following phenomenon observed inthis system. Two chaotic circuits coupled by a resistor havesame circuit structure. By changing a value of the coupling resistor, synchronization rate of a part of one circuit
is increased and the other is decreased. Namely, in spite of- 207 -Figure 7: Synchronous rate of Circuit A and Circuit B. αa1 = αa2 = αb1 = αb2 = 9, βa2 = 0.8, βb1 = 0.4, βb2 = 0.7, βa3 =βb3 = 0.32, γa2 = γb1 = γb2 = 1.0 and δa = δb = 1.4.
References[1] M. Shinriki, M. Yamamoto and S. Mori, “MultimodeOscillations in a Modified van der Pol Oscillator containing a positive nonlinear conductance,” Proc. IEEE,
vol. 69, no. 3, pp. 394-395, 1981.

synchronization[2] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena
in a Circuit with a Negative Resistance and an Ideal
Switch of Diodes,” IEICE Trans., vol. E-70, pp. 744–754, 1987.

Figure 6: Definition of synchronization.

being same circuit structure, an inverse result is obtained.
Additionally, one important point of this result is that twocircuits are not synchronized at all. In this condition, it isinteresting that these two circuits influence each other.

[3] H. Sekiya, S. Moro, S. Mori, and I. Sasase, “Synchronization Phenomena on Four Simple Chaotic Oscillators Full-Coupled by Capacitors,” IEICE Trans. Fundamentals, vol. J82-A, No. 3, pp.375–385, 1999.
[4] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in
Globally Coupled System of Chaotic Circuits,” Proc.
ISCAS’02, vol. 3, pp. 57–60, 2002.
[5] L. M. Pecora and T. L. Carrol, “Synchronization in
Chaotic Systems,” Physical Review Letters, vol. 64, pp.
821–824, 1990.

4. Conclusion
In this study, two modified Shinriki-Mori circuits coupled by a resistor are proposed. By investigating the relationship between subcircuits and a coupling strength, it isconfirmed that synchronization rate of a part of one circuit
is increased and the other is decreased by changing a coupling strength. We consider that the most important pointof this result is that two circuits are not synchronized at all
and two circuits influence each other.

[6] T. Sobue, Y. Hosokawa and Y. Nishio, “Coupled
Chaotic System Based on Shinriki-Mori Circuit,” Proc.
NCSP’14, pp. 529–532, 2014.

In the future works, we will investigate this phenomenon
in detail.

- 208 -