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

N-Phase Synchronization Observed in a Ring ofLarge-Scale Coupled Circuits
Takuya Nishimoto∗ , Yoko Uwate∗ , Yasuteru Hosokawa† , Yoshifumi Nishio∗ and Daniéle Fournier-Prunaret‡∗Dept. of Electrical and Electronic Eng., Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506 Japan
Email: {nishimoto, uwate, nishio}@ee.tokushima-u.ac.jp†
Dept. of Media and Information Systems, Shikoku University,Furukawa, Ohjin, Tokushima, 771-1192, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp‡LAAS-CNRS, 7 avenue du colonel Roche, F-31077 Toulouse Cedex 4, France
Email: daniele.fournier@insa-toulouse.fr
Abstract—Chaos synchronization can be observed in various
natural systems. In this study, we investigated synchronization
phenomena on a ring of coupled chaotic circuits. We can observeN-phase chaos synchronization and simple synchronization phenomena depending on initial values in the system. This paper
suggests ring topology and alternately shifted bias are essential
factor of N-phase synchronization.

I.

I NTRODUCTION

On the other hand, some chaotic circuit have coexisting attractors[1]. Asymmetric attractor observed in the system which is symmetric structure causes some interesting
phenomena[2][3]. However there are not many studies ofsynchronization on asymmetric attractor. It is considered thatobservation of synchronization phenomena in the state of chaos
on asymmetric attractor is difficult.
In this study, we also investigate synchronization phenomena in a ring of coupled chaotic system whose subcircuits generate asymmetric attractor. 2-types synchronization phenomena
depending on initial values can be stably observed in thesystem. Especially, N-phase chaos synchronization observed in
the system(N = 6 ∼ 50) is extremely interesting phenomena
since N-phase synchronization can be observed in small-scale
coupled chaotic system normally. In this study, we confirmed
N-phase synchronization phenomena on large-scale system.
By carrying out circuit computer simulations, we can confirm
the phenomena above. This paper is organized as follows. In
Section II, we propose the a ring of coupled chaotic system.
Section III, 2-type chaos synchronization are shown. Finally,
Section IV concludes this paper.
II.

S YSTEM MODEL

The system model is shown in Fig.1. Each subcircuit iscoupled by resister. Subcircuit is chaotic circuit which consists-1-R

Ri N1 i N2
L1

Chaos synchronization observed on large-scale coupledchaotic systems have been attracted attentions in various
science field because it can be regarded as models of real
physical system. For instance, chaos phenomena have beenreported in engineering, biology, economics, astronomy andso on. Therefore investigation of synchronization phenomena
on high-order chaotic systems are very important to grasp
essentials of the phenomena observed in natural system.

RR-r

L2vN

L1f(i N2)

-rFig. 1.

i 21 i 22

i 11 i 12L2v1

L1L2v2-rf(i 12)f(i 22)

System model.

three memory elements, one linear negative resister, and bidirectionally-coupled diodes proposed by Nishio et al[5]. First,
we approximate i − v characteristic of bi-directionally-coupled
diodes as following function:f (in2 ) =1(|rd in2 + V | − |rd in2 − V |)2(1)where parameter rd denotes slope of nonlinear resistance. Thenthe system equation is described by the following 3 × N dimensional ordinary differential equation:
din1

= vn + rin1 , L1 dt
din2= vn − f (in2 ),L2dt C dvn = −in1 − in2 + 1 (vn−1 − 2vn + vn+1 ) .
dt
R(2)
By changing the parameters and variables,√√√
CCt = L1 Cτ, in1 = V
xn , in2 = V
yn ,L1 √L1dC
L1(3)vn = V zn , ”· ” =,α = r,β =dτL2√ L1√
1 L1Cand δ =γ = rdL1R C

the normalized system equation is described as follows:ẋn = αx{n + zn ,} ẏn = β zn − fˆ(yn ) ,żn = −xn − yn+δ (zn−1 − 2zn + zn+1 ) ,(n = 1, 2, · · · , N )Zn(4)where(a)z0 = zN , zN +1 = z1 andfˆ(yn ) = 21 (|γyn + 1| − |γyn − 1|).
Figure 2 shows an example of asymmetric chaos attractor
depending on initial value obtained from one subcircuit. For
the computer simulation, we fixed the parameters as β = 3.0
and γ = 470.0. Asymmetric attractors are distinguished with
2 colors in simulation result. In this study, we distinguish theattractors as following definition:(1) A Poincaré section is defined at zn = 1.0 andẏn < 0.
(2) When the solution hits the Poincaré section, andyn > 0.675, the color is set as red and such stateis defined state {M }. In the case of yn ≤ 0.675,
the color is set as blue and defined state {P }.

Zn(b)Fig. 3. 2-type synchronization phenomena depending on initial values. (a)N-phase chaos synchronization. (b) Synchronization. Vertical: zn . Horizontal:
τ . N = 10, α = 0.425, β = 3.0, γ = 470.0 and δ = 0.20.

PThis definition is important to grasp the domain of attractor
in each subcircuit and as such, is being applied to following
simulation results in this paper.
III.

MM

2- TYPE CHAOS SYNCHRONIZATION

2-type synchronization phenomena depending on initialvalues can be observed in the system shown in Fig. 1. Figure 3shows the examples of synchronization phenomena. N-phasechaos synchronization is shown in Fig. 3 (a), and synchronization in Fig. 3 (b). Red and blue colors show the statePP or M
Fig. 4.

(a) α = 0.35(b) r = 4.99
Fig. 2. Asymmetric chaos attractor observed in a subcircuit. (a) Simulation
result. Horizontal: y1 . Vertical: z1 . β = 3.0 and γ = 470.0. (b) Circuit experiment. Horizontal: i12 [0.1mA/div]. Vertical: v1 [V/div]. L1 = 642.8[mH],L2 = 211.1[mH] C = 7.06[nF].

-2-Overview of attractor domain.

{M } and {P } respectively. Both synchronization states are
not synchronized rigorously since the system generates chaoticsignal. However, those behavior shown in Fig. 3 (a) and (b) arealmost synchronized. Therefore these phenomena are called
N-phase synchronization and synchronization respectively inthis paper. It should be noted that when the subcircuits (nis odd number) take state {P }, while the others (n is evennumber) take state {M } in Fig. 3. Namely, the subcircuits take
opposite attractor state with respect to the neighbor subcircuits.
An overview of attractor domain is shown Fig. 4. This is
the one of the characteristic of this chaotic circuit in case of
coupled by resister. It is considered that shifted bias waveformbe caused due to the characteristic in this system. Figure 5 issimulation results in state of N-phase chaos synchronization.
The phase difference between z1 and zn are gradually enlarged
from z1 vs z2 to z1 vs z6 . Moreover, z6 , which are locateddiagonal position of z1 , is almost synchronized with antiphase. Figure 6 shows time waveforms and lissajous figureof z1 and z6 observed in simulation. While z1 takes state{P }, z6 takes state {M } and almost synchronized with anti-IV.

(a) z1 vs z2(b) z1 vs z3(c) z1 vs z4

C ONCLUSION

In this study, we have investigated synchronization phenomena on coupled nonlinear circuit as a ring. As a result, Nphase synchronization and simple synchronization phenomena
are observed in a ring of large-scale coupled circuits. It isconsidered that essential factors of 2-type synchronization
depending on initial values are ring topology and alternately
shifted bias. This result is extremely interesting and would givethe effective suggestion to make clear the nonlinear phenomena
in large-scale high dimensional system. In our future work,
we investigate the mechanism of N-phase synchronization
phenomena on another circuit system.
R EFERENCES(d) z1 vs z5(e) z1 vs z6(f) z1 vs z7(g) z1 vs z8(h) z1 vs z9(i) z1 vs z10

Fig. 5. Lissajous figures in state of N-phase chaos synchronization (Computersimulations). z1 vs zn . α = 0.425, β = 3.0, γ = 470.0 and δ = 0.2.

Z1(a)
Z6

Z1(b)
Z6(1)(2)Fig. 6. Almost anti-phase synchronization state for N = 10. (a) N-phasesynchronization. (b) In-phase synchronization. (1) Time waveforms of z1 andz6 . (2) z1 vs z6 . (parameter values are the same as those in Fig. 5).

phase as shown in Fig. 6 (a). Figure 6 (b) shows another typesynchronization. Although z1 and z6 take opposite attractor
state, these circuits are almost synchronized with in-phase.
These behavior shown in Fig. 6 (a) and Fig. 6 (b) can beobserved by giving different initial values.
From these results, we can confirm that subcircuits locatedin diagonal position are synchronized with anti-phase or inphase though both subcircuits take opposite attractor state.
Moreover, 2-type synchronization phenomena can be observedin large-scale case around N = 50.

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