ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMSS EPTEMBER 17-20, 2006, Ł ÓD Ź , P OLAND

On Phase Synchronization of Simple Coupled
Chaotic Circuits
Koichi Matsumoto† , Yoko Uwate† and Yoshifumi Nishio††Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
TEL: +81-88-656-7470
FAX: +81-88-656-7471
E-mail: †{koichi, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract – The synchronization phenomena are therepresentative phenomena in the nonlinear science and canbe often observed in nature world. Coupled oscillatory
systems have attracted a great deal of attentions in various
fields. In particular, synchronization in such systems is very
important phenomenon and many researches have beenreported.
In this study, we investigate synchronization phenomena
in two coupled chaotic circuits. By computer simulations andcircuit experiments, interesting synchronization phenomena
are observed.
1. Introduction

2. Circuit ModelIn this study, we consider a simple coupled system of two
chaotic circuits to investigate the generation of the 143
degrees synchronization. The circuit model is shown in Fig.
1. In this model, two chaotic circuits are coupled by oneresistor R.

gC1


u1


id 1i1v
 1v
 2

C2C2

L

R

id 2

Li2


u2


C1

g

Coupled oscillatory systems have attracted a great deal ofattentions in various fields. Studies on synchronization
phenomena of coupled oscillators are extensively carried outFig. 1. Circuit model.
in various fields, physics [1]-[3], biology [4], engineering[5]-[12] and so on.
In our past studies [13][14], we have reported that a The characteristics of the nonlinear resistor consisting ofcertain class of coupled systems of the Wien-Bridge diodes are approximated by the following function.
oscillators synchronizes with 0 or 120 degrees.
di§·In [15], we have investigated synchronization phenomena
idk G¨ uk  L k  E ¸, (k 1,2).
in coupled Wien-Bridge oscillators by both circuitdt©
¹
experiments and computer simulations using SPICE. We
idk
observed the synchronization state with 143 degrees phasedifference. The phase difference of 143 degrees is very
strange, because in-phase, anti-phase, or N-phasesynchronization is typical.
GIn this study, phase synchronization phenomena in two
coupled chaotic circuits are investigated. By computer
diuk  L ksimulations and circuit experiments, we confirm that thedt0E
synchronization state with 143 degrees phase difference in
two coupled chaotic circuit is observed.
In Sec. 2, the circuit model in this study and the circuit
Fig. 2. The characteristics of the nonlinear resistor
equations are given. In Sec. 3, simulation results and circuit
consisting of diodes.
experiment are shown. In Sec. 4, we conclude our study.

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ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMS(iii) D0+The circuit equations are given as follows.

duk= guk − idkdtdv
C2 k = idk − ik
dtdik
= vk + R(id 1 − i1 + id 2 − i2 )
L
dt

⎛ δγ
⎞δγ (x2 − y2 − 1)− δ ( z1 + z2 )⎜⎜− 1⎟⎟ ≤ 1,1 + δγ
1δγ+⎝⎠
x2 − y2 − 1 δ ( z1 + z 2 )+>01 + δγ
1 + δγ

C1x1 − y1 −x&1 = αβ x1
y&1 = − z1

(k = 1, 2) .
By changing variables and parameters,``⋅ ' ' =d, t=dτLC1τ , uk = Exk , vk = Eyk , ik =z&1 = y1 +C2Ez k ,Lδγ(x2 − y2 − 1) + δ ( z1 + z2 )⎛⎜⎜ δγ − 1⎞⎟⎟
1 + δγ⎝ 1 + δγ⎠⎧γ
(x2 − y2 − 1) − δγ ( z1 + z2 )⎫⎬x&2 = α ⎨β x2 −1 + δγ
1 + δγ⎩⎭γ
δγ(x2 − y2 − 1) +y& 2 =( z1 + z 2 ) − z 2
1 + δγ1 + δγyδγ(x2 − 1) + δ ( z1 + z2 )z&2 = 2 +1 + δγ 1 + δγ
1 + δγCLLL
α= 2, β=g, γ =G,δ =R, (k = 1,2) ,C1
C2C2C2The circuit equations are normalized as follows.
(i) D00x k − y k + δ ( z1 + z 2 ) ≤ 1, ( k = 1, 2 ).

(iv) D++x&k = αβ xk
y& k = − z kz&k = yk − δ ( z1 + z 2 )δγ{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}1 + 2δγ−(k = 1, 2) .

δγ

−x1 − y1 − 1 δ ( z1 + z 2 )+> 0,1 + δγ
1 + δγ⎛ δγ⎞δγ (x1 − y1 − 1)+ δ ( z1 + z 2 )⎜⎜− 1⎟⎟ ≤ 1x2 − y 2 −+δγ
1 + δγ1⎝⎠x&1 = αβ x1 + αγ
− αγ
y&1 = γ⎧γ
(x1 − y1 − 1) − δγ ( z1 + z2 )⎫⎬x&1 = α ⎨βx1 −1 + δγ
1 + δγ⎩⎭γ
δγ(x1 − y1 − 1) +y&1 =( z1 + z 2 ) − z1
1 + δγ
1 + δγδγy(x1 − 1) + δ ( z1 + z2 )z&1 = 1 +1 + δγ 1 + δγ
1 + δγ

1 + δγ{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}+ (x2 − 1) > 0
1 + 2δγ

1 + δγ{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}1 + 2δγ

δγ
1 + 2δγ

δγ
1 + 2δγ

δγ(x1 − y1 − 1) + δ ( z1 + z2 )⎛⎜⎜ δγ − 1⎞⎟⎟
1 + δγ⎠⎝ 1 + δγ76{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}− γ (x1 − 1) − z1

1 + δγ{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}1 + 2δγ−x&2 = αβ x2
y& 2 = − z 2{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}− αγ (x1 − 1)

1 + δγ{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}1 + 2δγ−γ

z&1 =1 + δγ{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}+ (x1 − 1) > 0
1 + 2δγ{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}1 + 2δγ(ii) D+0z&2 = y2 +Ł ÓD Ź , P OLAND

δγ

1 + 2δγ{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMS

x&2 = αβx2 + αγ− αγy& 2 = γŁ ÓD Ź , P OLAND

1 + δγ{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}1 + 2δγ

δγ
1 + 2δγ{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}− αγ (x2 − 1)

1 + δγ{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}1 + 2δγ−γ

δγ
1 + 2δγ{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}− γ (x2 − 1) − z21 + δγ
{y2 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}z&2 =1 + 2δγ−

δγ

1 + 2δγ

Fig. 3 Computer simulation result. (a) Chaotic attractor. y1vs. x1. (b) Chaotic attractor. y2 vs. x2. (c) Phase difference.
x1 vs. x2. (d) Time waveform. α = 0.33, β = 0.43, γ =14.0, and δ = 0.05.

{y1 + δγ (x1 + x2 − 2) − δ ( z1 + z2 )}3. Simulation and Circuit Experiment Resultsv1

We investigate phase synchronization phenomena in two
coupled chaotic circuits by computer simulations using theRunge-Kutta method. Figure 3 shows the simulated results.
Figures 3(a) and (b) show the chaotic attractor of each
subcircuit. The phase difference and the time waveforms areshown in Figs. 3(c) and (d), respectively. From these figures,the two coupled chaotic circuits are synchronized witharound 143 degrees phase differences same as the two
coupled RC oscillators.
We also confirm the generation of the same
synchronization states in circuit experiments as shown inFig. 4.

(a )

(d )

(b )

(a )

u1(b )u2(c )

v1v2

v2

(c )
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ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMSŁ ÓD Ź , P OLAND[7] T. Endo and S. Mori, “Mode analysis of a multimodeladder oscillator,” IEEE Trans. Circuits Syst., vol.CAS23, no.2, pp100-113, Feb. 1976.

v1[8] T. Endo and S. Mori, “Mode analysis of a twodimensional low-pass multimode oscillator,” IEEE Trans.
Circuits Syst. vol.CAS-23, no.9, pp517-530, Sep. 1976.

v2(d )[9] S.P. Datardina and D.A. Linkens, “Multimodeoscillations in mutually coupled van der Pol typeoscillators with fifthpower nonlinear characteristics,”
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tFig.4 Circuit experimental result. (a) Chaotic attractor. u1
vs. v1. (b) Chaotic attractor. u2 vs. v2. (c) Phase difference.
v1 vs. v2. (d) Time waveform. C1 = 100nF, C2 = 33nF, L =100mH, and R = 100Ω.

[10] T. Endo and S. Mori, “Mode analysis of a ring of alarge number of mutually coupled van der Poloscillators,” IEEE Trans. Circuits Syst., vol.CAS-25,
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synchronization phenomena in two coupled chaotic circuits.
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“Wave propagation phenomena of phase states inoscillators coupled by inductors as a ladder,” IEICE
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