N-Phase Synchronization of Asymmetric Attractors
in a Ring of Coupled Chaotic 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 investigate synchronization
phenomena on a ring of coupled chaotic circuits. We can observe N-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
factors of N-phase synchronization.

I.

type chaos synchronization are shown. Be based on Section III,
another example with VDP model are explained in Section IV.
Finally, Section V concludes this paper.
II.

S YSTEM MODELI NTRODUCTION

Chaos synchronization observed on large-scale coupledchaotic systems have been attracted attention in various sciencefield because it can be regarded as models of real physicalsystem. For instance, chaos phenomena have been reported inengineering, biology, economics, astronomy and so on. Therefore the investigation of synchronization phenomena on highorder chaotic systems are very important to grasp essentials of
the phenomena observed in natural system.
On the other hand, some chaotic circuits 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 confirm
N-phase synchronization phenomena on large-scale system.
Moreover, similar behaviour can be observed in another systemwith VDP circuit which include ideal current source. By
carrying out circuit experiments and computer simulations, we
can confirm the phenomena above. Namely, the phenomenapossibly observed in many systems which have asymmetricattractor and ring topology.
This paper is organized as follows. In Section II, we
propose a ring of coupled chaotic system. In Section III, 2-

978-1-4799-3432-4/14/$31.00 ©2014 IEEE1780

RRRR
i N1 i N2
L1-r

L2vN

L1f(i N2)

-r

Fig. 1.

i 21 i 22

i 11 i 12L2v1

L1

L2v2

-r
f(i 12)f(i 22)

System model.

The system model is shown in Fig. 1. Each subcircuit iscoupled by resistor. Subcircuit is chaotic circuit which consistsof three memory elements, one linear negative resistor, 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
L2
= vn − f (in2 ),
dt


 C dvn = −in1 − in2 + 1 (vn−1 − 2vn + vn+1 ) .
dtR(2)

By changing the parameters and variables,√√√C
Ct = L1 Cτ, in1 = Vxn , in2 = Vyn ,L1 √L1dC
L1vn = V zn , ”· ” =,α = r,β =dτLL2√ 1√C1 L1γ = rdand δ =L1R 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 )

III.

(3)

(4)

wherez0 = zN , zN +1 = z1 andfˆ(yn ) = 12 (|γyn + 1| − |γyn − 1|).

2- TYPE CHAOS SYNCHRONIZATION

2-type synchronization phenomena depending on initialvalues can be observed in the system shown in Fig. 1. Figure 3
shows 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 state{M } and {P } respectively. Both synchronization states are
not synchronized rigorously since the system generates chaotic
signal. However, those behaviours shown in Fig. 3 (a) and (b)are almost 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 in Fig. 4. This is
the one of the characteristic of this chaotic circuit in case of
coupled by resistor. It is considered that shifted bias waveformbe caused due to the characteristic in this system. Figure 5 isFigure 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 follows:Zn(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 }.

(a)This 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.

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.

P(a) α = 0.35

MMP(b) r = 4.99

P or M

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].

1781

Fig. 4.

Overview of attractor domain.

(a) z1 vs z2(b) z1 vs z3(c) z1 vs z4(a) v1 vs v2(b) v1 vs v3(c) v1 vs v4(d) z1 vs z5(e) z1 vs z6(f) z1 vs z7(d) v1 vs v5(e) v1 vs v6(f) v1 vs v7(g) z1 vs z8(h) z1 vs z9(i) z1 vs z10(g) v1 vs v8(h) v1 vs v9(i) v1 vs v10

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.

simulation results in state of N-phase chaos synchronization.
The phase difference between z1 and zn is enlarged from z1 vsz2 to z1 vs z6 . Moreover, the subcircuits of diagonal positions
are almost synchronized with anti-phase. This behaviour also
can be observed in circuit experiment as shown in Fig. 6.
Figure 7 shows time waveforms and Lissajous figure of z1
and z6 observed in simulation and circuit experiment. While z1
takes state {P }, z6 takes state {M } and almost synchronizedwith anti-phase as shown in Fig. 7. Figure 8 shows another typesynchronization. Although z1 and z6 take opposite attractor
state, these circuits are almost synchronized in-phase. Thisbehaviour can be observed by giving different initial values.
From this result, we can confirm that subcircuits which arelocated in diagonal position are synchronized with anti-phaseor in-phase though both subcircuits take opposite attractor
state. Moreover, 2-type synchronization phenomena can beobserved in large-scale case around N = 50.
IV.

A NOTHER EXAMPLE

From Section III, we found that two-type synchronization
state is caused by ring topology and shifted bias waveforms.
Since asymmetric attractors can be regarded as alternately
shifted biases. Therefore we propose new subcircuit modelwith VDP circuit which includes ideal current source in order
to confirm the essential factors of 2-type synchronization. The
circuit model is shown in Fig. 9. Ideal current sources areused at the subcircuit located at n (odd number) with forward
direction, and the subcircuit located at k (even number) withbackward direction to make biases shift. We approximate the1782

Fig. 6. Lissajous figures in state of N-phase chaos synchronization (circuitexperiments). v1 vs vn [10V/div]. L1 = 625[mH] ± 1%, L2 = 209[mH] ±1%, C = 1.49[nF] ± 4% and R = 100.0[kΩ] ± 1%.

Z1(a)
Z6

Z1(b)
Z6(1)

(2)

Fig. 7. Almost anti-phase synchronization state for N = 10. (a) Simulation
result. (1) Time waveforms of z1 and z6 . (2) z1 vs z6 . (parameter values are
the same as those in Fig. 5). (b) Circuit experiment. (1) Time waveforms ofv1 and v6 . (2) v1 vs v6 . (parameter values are the same as those in Fig. 6).

vn − iRn characteristics of nonlinear resistor including ideal
current source as follows:

iRn = −g1 vn + g3 vn3 + (−1)n J

(5)

Z1

(a)

(a)

Yn(b)

Yn

Z6

Z1(b)
Z6

(1)

(1)

(2)

Fig. 8. Synchronization state for N = 10. (a) Simulation result. (1) Time
waveforms of z1 and z6 . (2) z1 vs z6 . (parameter values are the same as
those in Fig. 5). (b) Circuit experiment. (1) Time waveforms of v1 and v6 .
(2) v1 vs v6 . (parameter values are the same as those in Fig. 6).

RRi RN i LN

From these results, we specified that the essential factorsof 2-type synchronization are ring topology and alternately
shifted biases.

i R2 i L2
i R1 i L1vN Lv2
v1

JLV.

L JJFig. 9.

System model with VDP-model.

By changing the parameters and variables,
√ √√
√g1
g1 C
t = LCτ, vn =xn , iLn =yn ,gg3 L√√3√ √Lg3 L1 L
ε = g1, δ=and β =JCR C
g1 C
the normalized system equation is described as follows:2n ẋn = ϵ(1 − xn )xn − yn + (−1) β+δ(xn−1 − 2xn + xn+1 ), ẏn = xn ,(n = 1, 2, · · · , N )Fig. 10. 2-type synchronization phenomena observed in another system model
as shown in Fig. 9. (a) N-phase synchronization state. (b) Synchronization
state. (1) Time waveforms of yn . (2) y1 vs y6 . N = 10, ε = 0.08, β = 0.3
and δ = 0.05.

In this case, each subcircuit exactly synchronized with Nphase. Moreover, the subcircuits located in diagonal position
are synchronized with opposite phase shown in Fig. 10 (2).
Those 2-type synchronization phenomena also can be observed
around N = 50.

RR

(2)

(6)

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 2 system models. Moreover, it is declared thatessential factors of 2-type synchronization depending on initialvalues are ring topology and alternately shifted bias. This resultis extremely interesting and would give the effective suggestion
to make clear the nonlinear phenomena in large-scale highdimensional system. In our future work, we will investigate
the mechanism of N-phase synchronization phenomena in thesystem with VDP model by using averaging method.
R EFERENCES(7)

wherex0 = xN and xN +1 = x1 .
Figure 10 shows simulation results for the system shownin Fig. 9. 2-type synchronization depending on initial values
also can be observed in this system. The time waveforms aredescribed with 2 colors following condition:(1) Blue: n is odd.
(2) Red: n is even.

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