2009 International Symposium on Nonlinear Theory and its Applications
NOLTA'09, Sapporo, Japan, October 18-21, 2009

Reflection Mechanisms of a Phase-Inversion Wave at an Edge of 2D
Lattice Oscillators
Hitoshi ABURATANI† , 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‡Department of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: ab.hitoshi247@gmail.com
Abstract—We investigate synchronization phenomena on coupled oscillators system that van der
Pol oscillators are coupled by inductors as a lattice.
In the lattice oscillators, the phase-inversion waves,which are phenomena changing phase states between
adjacent oscillators from in-phase synchronous to antiphase synchronous or from anti-phase synchronous toin-phase synchronous and exists in steady state, werediscovered. This paper clariﬁes reﬂection mechanismsof a phase-inversion wave at the edge by using instantaneous frequencies of each oscillator and phase diﬀerences between adjacent oscillators in the lattice shapesystem.
1. Introduction
Many investigations and analyses for synchronous
systems are carried out up to now[1]–[4]. A lot ofkinds of synchronization phenomena exists anywhere
and anytime. For example, there are synchronous
between sea waves and ship motion, earthquake andbuildings, and so on. Therefore, synchronization phenomena may bring on disaster. We think that these
problems of the natural world may be clariﬁed by analyzing many synchronization phenomena. Moreover,
we think that the time-series data of natural world
become to be able to predict by using a synchronization system. By investigating the synchronization phenomena of coupled oscillators, we believe that various
synchronization phenomena can be modeled using theelectrical circuits. We observe the various phenomena on coupled oscillators system, and analyze thesemechanisms[5].
In our previous study, we investigated phenomena insystems that many van der Pol oscillators were coupled
as a lattice shape. The coupling parameter and thenonlinearity of the system were changed, and the existing region of phase-inversion waves were made clear[6].
Moreover, we developed the prediction system of thetime-series data using the lattice oscillators. The chaostime-series data of Inaba’s circuit and Chua’s circuit
were predicted by this system[7].
In this paper, we analyzes two kinds of reﬂection

(1,1)

L0

OSC

(1,2)
OSC

L0

L0

(1,N)
OSC

L0
(2,2)
(2,1)

OSC

OSC

OSC

i(k,l)

L0

OSC

L C

v(k,l)

L0
(M,1)

OSC

OSC

OSC

(M,N)

Figure 1: Circuit Model.
phenomena of the phase-inversion waves using instantaneous frequencies of each oscillator and phase diﬀerences between adjacent oscillators at an edge in thelattice shape system. The mechanisms of each reﬂection phenomenon are made clear.
2. Circuit model
A lot of van der Pol oscillators are coupled by inductors L0 as a lattice(see Fig. 1). The number of columnof the system is assumed as M . The number of row of
the system is assumed as N . The name of each oscillator are assumed to be OSC(k,l). A voltage of each
oscillator is named v(k,l) , and a current of inductor ofeach oscillator is named i(k,l) (see Fig. 1). The circuit
equations of the circuit model are normalized by Eq.
(1), and the normalized circuit equations are shown as
Eqs. (2)–(6).


Cg1
g1
x
,
v
=
i(k,l) = 3Lg
(k,l)
(k,l)
3g3 y(k,l) ,
3

(1)
√
d
L
= “ · ”, α = LL0 , ε = g1 C.
t = LCτ, dτ[Corner–top (left and right)]
dx(1,a)= y(1,a) ,dτ(2)dy(1,a)= −x(1,a) + α(x(1,b) + x(2,a) − 2x(1,a) )dτ

1 3+ ε(y(1,a) − y(1,a)),
3
left:a = 1 and b = 2. right:a = N and b = N − 1.

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15% 

αβ15% 

[Corner–bottom (left and right)]
dx(M,a)= y(M,a) ,dτ(3)dy(M,a)= −x(M,a) + α(x(M−1,a) + x(M,b)
dτ3− 2x(M,a) ) + ε(y(M,a) − 13 y(M,a)),
15% 

left:a = 1 and b = 2. right:a = N and b = N − 1.
[Center]dx(k,l)= y(k,l) ,dτ(4)dy(k,l)= −x(k,l) + α(x(k+1,l) + x(k−1,l) + x(k,l+1)dτ
3+ x(k,l−1) − 4x(k,l) ) + ε(y(k,l) − 13 y(k,l)),

1 < k < M. 1 < l < N.
[Edge–top and bottom]
dx(a,l)= y(a,l) ,dτ(5)dy(a,l)= −x(a,l) + α(x(a,l−1) + x(a,l+1) + x(b,l)
dτ
3− 3x(a,l) ) + ε(y(a,l) − 13 y(a,l)),

top:a = 1 and b = 2. bottom:a = M and b = M − 1.
both:1 < l < N .
[Edge–left and right]dx(k,a)= y(k,a) ,dτ(6)dy(k,a)= −x(k,a) + α(x(k−1,a) + x(k+1,a) + x(k,b)
dτ
3− 3x(k,a) ) + ε(y(k,a) − 13 y(k,a)),

left:a = 1 and b = 2. right:a = N and b = N − 1.
both:1 < k < M .
The α is a coupling parameter of each oscillator. The
ε is a nonlinearity of each oscillator. This system issimulated by the fourth order Runge-Kutta methods
using Eqs. (2)-(6).
3. Reflection mechanisms at an edge
The phase-inversion waves shows in Fig.2. The reﬂection mechanisms at an edge are made clear using
instantaneous frequency of each oscillator and phase
diﬀerence between adjacent oscillators. The coupling
parameter is ﬁxed as α = 0.01, and nonlinearity is
ﬁxed ε = 0.050. An equation of the instantaneous
frequency of OSC(k, l) is calculated as follows. The
instantaneous frequency is named f(k,l) (a) where “α”
expresses the number of times of the peak value ofthe voltage. Time of a peak value of the voltage ofOSC(k, l) is assumed as τ(k,l) (a)(see Fig.3). Similarly,
τ(k+1,l) (a) and τ(k,l+1) (a) are decided. The f(k,l) (a) is
obtained by Eq.(7).

15% 

Figure 2: The phase-inversion waves on 20×20
oscillators(α:an attractor of each oscillator(current
vs. voltage), β:a sum of voltages of adjacent oscillators(sum of voltage vs. time)).
Three frequencies are observed in steady states. In
this system, the synchronous for vertical direction andhorizontal direction needs to be considered, because
this system is 2 dimensional array. Therefore, three
type synchronizations are observed as follows:
1. OSC(k, l)–OSC(k, l + 1), OSC(k, l)–OSC(k, l − 1),
OSC(k, l)–OSC(k + 1, l), and OSC(k, l)–OSC(k − 1,l): in-phase synchronous.
2. {OSC(k, l)–OSC(k, l + 1) and OSC(k, l)–OSC(k,l − 1): in-phase synchronous. OSC(k, l)–OSC(k + 1,l), and OSC(k, l)–OSC(k − 1, l): anti-phase synchronous.} or {OSC(k, l)–OSC(k, l + 1), and OSC(k,l)–OSC(k, l − 1): anti-phase synchronous. OSC(k, l)–OSC(k + 1, l), and OSC(k, l)–OSC(k + 1, l): in-phasesynchronous.}
3. OSC(k, l)–OSC(k, l + 1), OSC(k, l)–OSC(k, l − 1),
OSC(k, l)–OSC(k + 1, l), and OSC(k, l)–OSC(k + 1,l): anti-phase synchronous.
In this paper, we call the 1st type synchronous “inand-in-phase synchronous.” The 2nd type synchronization is called “in-and-anti-phase synchronous.”
The 3rd type synchronization is called “anti-and-antiphase synchronous.” An each instantaneous frequency
of OSC(k, l) is obtained in each synchronous type. In
the 1st situational synchronous, f(k,l) is fin−in . In the
2nd situational synchronous, f(k,l) is fin−anti . In the
3rd situational synchronous, f(k,l) is fanti−anti .
The phase diﬀerence is calculated as follows. A
phase diﬀerence between OSC(k, l) and OSC(k + 1, l)
and a phase diﬀerence between OSC(k, l) and OSC(k,l + 1) are calculated. The phase diﬀerences are assumed as Φ(k,l)(k+1,l) (a) and Φ(k,l)(k,l+1) (a) respectively.The Φ(k,l)(k+1,l) (a) and Φ(k,l)(k,l+1) (a) are obtained by Eq.(8).
τ(k,l) (a) − τ(k+1,l) (a)× 180 [degree]τ(k,l) (a) − τ(k,l) (a − 1)(8)
τ(k,l) (a) − τ(k,l+1) (a)Φ(k,l)(k,l+1) (a) =× 180 [degree].
τ(k,l) (a) − τ(k,l) (a − 1)Φ(k,l)(k+1,l) (a) =τ(k, l)(a)V

OSC(k, l)

a-1

a-2

a+1

a+2

a+3

τ(k+1, l)(a)V

1.
f(k,l) (a) =τ(k,l) (a) − τ(k,l) (a − 1)

a

Time

OSC(k+1, l)
a-1

(7)

a-2

a

a+1

a+2

a+3

Time

Figure 3: The detection method of frequencies and thephase diﬀerences.

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Table 1: Reﬂection mechnism of a phase-inversion
wave at an edge in in-and-anti phase synchronous(seeFig.4).
0

Instantaneous Frequency

no.

fanti-anti

Mechanism
The phase state between adjacent oscillators of thehorizontal direction is the in-phase synchronous.

f (2,48)f (1,48)
8

fin-in1

f (0,48)

5

3

A phase-inversion wave, which propagates from
OSC(9,48) in in-phase synchronous, arrives at

13

11

Time[τ](a)Instantaneous frequency

12

14

360

OSC(2,48).

Φ(0,48)(1,48)

Phase Difference[degree]

1

9

7

fin-antif(2,48) starts to increase from fin−in toward fin−antiby the phase-inversion wave. f(2,48) can not arriveat fanti−anti because phase states of horizontal directions of these oscillators are in-phase synchronous.

6

180

8 10

Φ(1,48)(2,48)
4

2
0

2

Φ(1,48)(2,48) starts to increase toward 180 degrees from0 because f(2,48) increases toward fin−anti .

3
4

f(1,48) starts to increase from fin−in toward fin−antibecause Φ(1,48)(2,48) increases toward 180 degrees.

f(0,48) starts to increase toward fin−anti because
Φ(0,48)(1,48) increases toward 180 degrees.

6Φ(1,48)(2,48) arrives around 180 degrees, and is stablilized as the anti-phase synchronous.

7

f(1,48) arrives around fin−anti .

8f(1,48) arrives at fin−anti , and Φ(0,48)(1,48) becomesaround 180 degrees.
However, in these parameters, OSC(0,48) can not stablilize for vertical direction. Therefore, f(0,48) becomes middle of fin−in andfin−anti , and decreases to fin−in again.

9

f(1,48) starts to decrease from fin−anti toward fin−inbecause Φ(0,48)(1,48) increases toward 360 degrees.

10

Figure 4: Transitions of phase diﬀerences and frequencies by reﬂecting a phase-inversion wave in in-and-inphase synchronous.

Φ(0,48)(1,48) starts to increase toward 180 degrees from0, because f(1,48) increases toward fin−anti .

5Time[τ ](b)Phase differenceΦ(1,48)(2,48) starts to increase toward 360 degrees from180 because f(1,48) decreases toward fin−in.

11

f(0,48) arrives around fin−in , and becomes stable.

12Φ(0,48)(1,48) arrives around 360 degrees and becomes

In Fig.4(a), the vertical axis is the instantaneous frequency, and the horizontal axis is time. In Fig.4(b),the vertical axis expresses the phase diﬀerence and thehorizontal axis expresses time.
3.2. Reflection of phase-inversion waves in inand-anti-phase synchronous
We can observe that a phase-inversion wave fora vertical direction in each column propagate in inand-anti-phase synchronous. A mechanism is shownas Tab.2. A mechanism of a phase-inversion wave
for other edges can be explained same method. In
Fig.5(a), the vertical axis is the instantaneous frequency, and the horizontal axis is time. In Fig.5(b),the vertical axis expresses the phase diﬀerence and thehorizontal axis expresses time.

14

Instantaneous Frequency

stable.
13

6

fanti-antiΦ(0,48)(1,48) arrives around 360 degrees, and f(1,48)
arrives around fin−in and becomes stable.
f(1,48) arrives around fin−in , and Φ(1,48)(2,48) arrives
around 360 degrees and becomes stable.

89f(1,8)f (2,8)f(0,8)

fin-anti
1

3

10

12

14

fin-inTime[τ ](a)Instantaneous frequency

The phase-inversion waves are generated by the following method. 1. Voltages and currents of all oscillators are set as same value, in other words all oscillators
are set as in-phase synchronization. 2. The signs ofx(k,l) and y(k,l) of arbitrary oscillators are inverted instantaneously.
3.1. Reflection of phase-inversion waves in inand-in-phase synchronous
We can observe that a phase-inversion wave for a
vertical direction in each column propagate in in-andanti-phase synchronous. A mechanism is shown as
Tab.1. A mechanism of a phase-inversion wave fora horizontal direction can be explained same method.

11

Phase Difference[degree]

360

13

Φ(0,8)(1,8)

5

180

7

Φ(1,8)(2,8)

0

2

4

Time[τ ](b)Phase difference

Figure 5: Transitions of phase diﬀerences and frequencies by reﬂecting a phase-inversion wave in in-and-antiphase synchronous at an edge.

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Table 2: Reﬂection of a phase-inversion wave at anedge in in-and-in-phase synchronous(see Fig.5).
no.

Mechanism0

The phase state between adjacent oscillators of thehorizontal direction is the anti-phase synchronous.
A phase-inversion wave,f(2,8)

Acknowledgements

which propagates from

OSC(9,8) in anti-phase synchronous,
OSC(2,8).
1

oscillator.
At the corner of our system, a reﬂection of the phaseinversion wave can be observed. We should make clear
the mechanism of the reﬂection.

arrives atstarts to increase from fin−antitowardThis research is supported by the Grants-in-Aid forYoung Scientiﬁc Research (B) (No. 19760270) fromthe Japan Society for the Promotion of Science.

fanti−anti by the phase-inversion wave.
2

Φ(1,8)(2,8) starts to increase toward 180 degrees from0, because f(2,8) increases toward fanti−anti .

3

f(1,8)

starts to increase from fin−antitowardfanti−anti because Φ(1,8)(2,8) increases toward 180 degrees.
4

Φ(0,8)(1,8) starts to increase toward 180 degrees from0, because f(1,8) increases toward fanti−anti .

5

Φ(1,8)(2,8) arrives around 180 degrees, and it stabilizes
the anti phase synchronous.

6
7

f(1,8) arrives fanti−anti because Φ(1,8)(2,8) arrived
180 degrees.
Φ(0,8)(1,8) is not steady in 180 degrees, and Φ(0,8)(1,8)
increases toward 360 degrees,because f(1,8) arrived atfanti−anti . f(0,8) can not arrive at fanti−anti . f(0,8)
arrives middle fanti−anti and fin−anti , and decrease
to fin−anti again.

7.1

Φ(0,8)(1,8) keeps passing 180 degrees, and it increasing
toward 360 degrees.

8Φ(0,8)(1,8) does not stop 180 degrees, and Φ(0,8)(1,8)
continues to change to 360 degrees. Therefore, f(1,8)
starts to decrease from fanti−anti to fin−anti .

9f(2,8) starts to decrease from fanti−anti toward
fin−anti because Φ(1,8)(2,8) increases toward 360 degrees from 180degrees.

10

f(0,8) arrives around fin−anti .

11

Φ(0,8)(1,8) arrives 360 degrees because f(0,8) arrived

12

f(1,8) arrives fin−anti because Φ(0,8)(1,8) arrived 360

13

Φ(1,8)(2,8) arrives 360 degrees because f(1,8) arrived

fin−anti .
degrees.
fin−anti .
14f(2,8) arrives fin−anti because Φ(1,8)(2,8) arrived 360
degrees.

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4. Conclusion
We can observe reﬂecting phase-inversion waves atthe edge in lattice oscillators. The reﬂections can beobserved at the edge of lattice oscillators in in-andin phase synchronous and in in-and-anti phase synchronous. These mechanisms of reﬂection phenomena
were made clear by phase diﬀerences between adjacent oscillators and instantaneous frequencies of each

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