2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009Classification of phenomena on coupled oscillators system as a lattice
Suguru Yamane† , Masayuki Yamauchi† and Yoshifumi Nishio‡†Department of Electronics and Computer Engineering,
Hiroshima Institute of Technology
2-1-1, Miyake, Saeki-ku, Hiroshima , Japan
Email: aqua blaster com@hotmail.co.jp‡Intelligent Networks and Computer Sciences,
Department of Electrical and Electronic Engineering,
Tokushima University(0,0)

Abstract

OSC

L0(0,1)OSC

L0L0(0,9)
OSC

L0In our previous study, we observed that phenomena of chang(1,1)
ing phase states between two adjacent oscillators from inOSC
OSC(1,0) OSC
phase to anti-phase or from anti-phase to in-phase and conL0
OSC
tinuously exist. We call the phenomenon “phase-inversion
waves.” The phase-inversion waves were observed on a ladL0
der of coupled oscillators. In this study, the phase-inversion
waves are observed on a lattice of coupled oscillators. TheOSC
OSC (9,9)(9,0) OSC
various phenomena of propagating phase diﬀerences, which
include phase-inversion waves, are observed by changing iniFigure 1: Circuit model.
tial values of each oscillator, coupling parameter and a nonlinearity of the circuit. Then, we clarify regions of the ob- waves are made clear.
served the phase-inversion waves.
2. Circuit model

1. Introductioni(k,l)
L Cv(k,l)

The van der Pol oscillators are coupled by inductors L0 asa lattice(see Fig.1). The number of rows of the system is 10.
The number of columns of the system is 10. Each oscillator
is named as OSC(k,l), a voltage of each oscillator is calledv(k,l) . i(k,l) is the current flowing through the inductor of eachoscillator(see Fig.1). An equation of the nonlinear negative
resistor is shown as Eq. (1). The circuit equations are normalized by Eq. (2). The normalized circuit equations are shownas Eqs. (3)–(7).

The studies of synchronization phenomena are researchedin physics, biology, and various fields[1]-[2]. For example,
the phase synchronization phenomena of the nervous system
are researched. The coupled oscillators system is known thatoscillators synchronize[3]. We investigate the synchronization phenomena by coupled oscillators system.
The phase-inversion waves, which are propagate in in-andanti-phase synchronization and in in-phase synchronization,
are observed on a ladder of coupled oscillators system[4].
f(k,l) = −g1 v(k,l) + g3 v3(k,l) .
In our previous study, we made a prediction system of
Cg1
g1
time-series data[5]. The system is that some van der Pol osi(k,l) =x(k,l) , v(k,l) =y(k,l) ,3Lg
3g
cillators are coupled by inductors as a lattice. We made clear
33√that the system can predict time-series data, but the system
LdL= “ · ”, α = , ε = g1.
t = LCτ,does not predict automatically. Therefore, phenomena on thedτL0C
system shall be analyzed for developing automatic prediction[Corner–top: left and right]
system.
In this study, we investigate the synchronization phenomdx(0,a)= y(0,a) ,ena on a lattice-shaped coupled van der Pol oscillators sysdτ
tem. The phase-inversion waves of vertical and horizontal didy(0,a)
rection are observed when initial values of voltages and cur= −x(0,a) + α(x(0,b) + x(1,a) − 2x(0,a) )dτ
rents of oscillators are set as a plus value or a minus value
1
alternately. State of phase-inversion waves are able to be+ ε(y(0,a) − y3(0,a) ).
3
observed three types when we change pattern of initial values. The observation regions of each kind of phase-inversion left:a = 0 and b = 1. right:a = 9 and b = 8.

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(1)

(2)

(3)

[Corner–bottom: left and right]dx(9,a)= y(9,a) ,dτ(4)dy(9,a)= −x(9,a) + α(x(8,a) + x(9,b) − 2x(9,a) )dτ
1+ ε(y(9,a) − y3(9,a) ).
3(5)(b) Initial values of pattern B.

Initial values are fixed to two patterns and the phaseinversion waves are observed. We observe the phaseinversion waves under the following observation conditions.
<Pattern A>
Observation conditions of Pattern A are fixed as follows.
1. Initial values are fixed as shown in Fig 2(a).
2. The coupling parameter α and nonlinearity ε arechanged.

where 0 < k < 9, 0 < l < 9.
[Edge–top and bottom](6)<Pattern B>
Observation conditions of Pattern B are fixed as follows.
1. Initial values are fixed as shown in Fig 2(b).

dy(a,l)= −x(a,l) + α(x(a,l−1) + x(a,l+1) + x(b,l) − 3x(a,l) )dτ
1+ ε(y(a,l) − y3(a,l) ).
3

2. The coupling parameter α and nonlinearity ε arechanged.
3.2. Experiment result

top:a = 0 and b = 1.
bottom:a = 9 and b = 8. both:0 < l < 9.
[Edge–left and right]dx(k,a)= y(k,a) ,dτ(a) Initial values of pattern A.

3.1. Observation conditions

dy(k,l)= −x(k,l) + α(x(k+1,l) + x(k−1,l) + x(k,l+1)dτ
1+ x(k,l−1) − 4x(k,l) ) + ε(y(k,l) − y3(k,l) ).
3dx(a,l)= y(a,l) ,dτ++−−++−−++++−−++−−++−−++−−++−−−−++−−++−−++−−++−−++++−−++−−++−−++−−++−−−−++−−++−−++−−++−−++++−−++−−++

Figure 2: Initial value.

left:a = 0 and b = 1. right:a = 9 and b = 8.
[Center]dx(k,l)= y(k,l) ,dτ++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−++−−(7)

dy(k,a)= −x(k,a) + α(x(k−1,a) + x(k+1,a) + x(k,b) − 3x(k,a) )dτ
1+ ε(y(k,a) − y3(k,a) ).
3
left:a = 0 and b = 1.
right:a = 9 and b = 8. both:0 < k < 9.
The α corresponds to the coupling of the oscillators. The εcorresponds to nonlinearity of each oscillator. These equations are calculated by the fourth order Runge-Kutta methods.
3. Classification of the phase-inversion waves
The phase-inversion waves on a lattice of coupled oscillators are observed and are classified some phenomena. The
initial values are changed to two patterns(A and B in Fig. 2),
and the phenomena are observe. Figure 2 shows the signs of
the initial values of the voltage and current of each oscillator.

The Figs.3 and 6 show the observation phenomena. TheFig.3(a)-(I) shows the attractor of each circuit. The horizontal axis expresses voltage y(n,m) , and the vertical axis expresses current x(n,m) . The Fig.3(a)-(II) shows a transition of aphase diﬀerence between adjacent oscillators. The horizontal
axis expresses time, and the vertical axis expresses a voltage
which is sum of two voltages of adjacent oscillators. The vertical axis is y(n,m) + y(n+1,m) or y(n,m) + y(n,m+1) . The amplitude
doubles if in-phase synchronization is observed between adjacent oscillators. On the other hand, the amplitude becomeszero if anti-phase synchronization can be observed between
adjacent oscillators. The Figs.5 and 7 show the region of
the observation phenomena. The vertical axis is nonlinearity. The horizontal axis is coupling parameter.
States, that the phase-inversion wave propagate, are classified to three types. First type wave of phase-inversion waves
is observed in in-and-anti-phase synchronization, and thephase-inversion waves propagate to vertical direction. Second type wave of phase-inversion waves is observed in inphase synchronization, and the phase-inversion waves propagate to horizontal direction. Third type wave of phaseinversion waves is observed in in-and-anti-phase synchronization, and the phase-inversion waves propagate to horizontal direction.

- 250 -(I)(II)(a) Phase-inversion waves of region-(i)(α=0.01 and ε=0.05).

Figure 6: Phase-inversion waves of region-(iv)(α=0.02 andε=0.15).
0.3

0.25

(iii)

ε0.2

(iv)

0.15

0.10.05(b) Phase-inversion waves of region-(ii)(α=0.02 and ε=0.02).

00Figure 3: Phenomena of phase-inversion waves.

0.010.02α0.03

0.04

0.05

Figure 7: Region of pattern B. Region-(iii): The complex
phenomena. Region-(iv): There are the first and the thirdtypes.

3.2.1. Pattern A

εObserved phenomena are classified into two patterns. Observation results are shown in Figs. 3 and 4, and are classifiedto Fig. 5.
Two types of states of the phase-inversion waves are observed.
In the Fig. 3(a), the first and the second type wave
Figure 4: An example of region-(iii)(pattern A, α=0.01 andareobserved(seeFig. 5-(i)). In the Fig. 3(b), the first typeε=0.02).
wave can be observed, but the second type wave can not be
0.1observed(see Fig. 5-(ii)).
0.09
The complex phenomena are observed outside the region
0.08
wherethe phase-inversion waves are observed(see Fig. 4 and(iii)0.07
5-(iii)).
0.06
0.05
0.04

(i)
(ii)

0.03

3.2.2. Pattern B

0.02

Observation result is shown in Fig. 6. Figure 7 shows theregions of the observation phenomena. In the Fig. 6, the firstα
and the third type waves are observed(see Fig. 7-(iv)).
The complex phenomena are observed outside the region
Figure 5: Region of pattern A. Region-(i): There are the firstwhere the phase-inversion waves are observed(see Fig. 4 andand the second types. Region-(ii): There are the first type.
7-(iii)).
Region-(iii): The complex phenomena.
0.01
00

0.01

0.02

0.03

0.04

0.05

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3.3. Specification of phenomenay(0,0)+y(1,0)
y(1,0)+y(2,0)
y(2,0)+y(3,0)
y(3,0)+y(4,0)
y(4,0)+y(5,0)
y(5,0)+y(6,0)
y(6,0)+y(7,0)
y(7,0)+y(8,0)
y(8,0)+y(9,0)

Figures 8 and 9 are extracted from the Fig. 3(a). The phaseinversion waves propagate in in-and-anti-phase synchronization and in-phase synchronization.
Figure 10 show the phase-inversion waves in the state ofin-and-anti-phase synchronization on a ladder of coupled oscillators. When we think about the vertical direction and horizontal direction separately, the same phenomena are observedon the ladder system and the lattice system.
4. Conclusion

Figure 8: Extraction of the Fig. 3(a) in vertical direction.

y(3,0)+y(3,1)
y(3,1)+y(3,2)
y(3,2)+y(3,3)
y(3,3)+y(3,4)
y(3,4)+y(3,5)
y(3,5)+y(3,6)
y(3,6)+y(3,7)
y(3,7)+y(3,8)
y(3,8)+y(3,9)

We observed and classified the phenomena of the phaseinversion waves. Observed phenomena were classified into
two patterns when initial values were fixed as Fig. 2(a). Observed phenomena were clarified when initial values werefixed as Fig. 2(b). On the ladder shape systems, we can observe phase-inversion waves which propagate in in-and-antiphase synchronization and in in-phase synchronization. On
the lattice shape systems, we can observe both types of propagation of phase-inversion waves at the same time.
Acknowledgements
This research is supported by the Grants-in-Aid for Young
Scientific Research (B) (No. 19760270) from the Japan Society for the Promotion of Science.
References

Figure 9: Extraction of the Fig. 3(a) in horizontal direction.

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Figure 10: Phase-inversion waves in the state of in-and-anti- [5] T. Imoto, K. Ichiki, S. Yamane and M. Yamauchi, “Phasephase synchronization on a ladder of coupled oscillators sysDiﬀerence Propagation Phenomena on a Non-edge Lattem.
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