Propagation Mechanism of Phase-Inversion Wave in
2D Lattice Oscillators and their Application to
Prediction of Time-Series Data
Hitoshi ABURATANI, Suguru YAMANE,
Takeo IMOTO and Masayuki YAMAUCHI

Yoshifumi NISHIO
Department of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan

Department of Electronics and Computer Engineering,
Hiroshima Institute of Technology
2-1-1, Miyake, Saeki-ku, Hiroshima , Japan
Email: ab.hitoshi247@gmail.com
Abstract— In our previous study, we discovered phaseinversion waves on the lattice shape system. The phase-inversion
waves are phenomena changing phase states between adjacentoscillators from in-phase to anti-phase or from anti-phase toin-phase and continuously. The phase-inversion waves can beobserved in steady states. In this study, we observe and analyze
propagation mechanisms of the phase-inversion waves in thelattice shape system. Furthermore, we predict air temperature
data by using the lattice shape system.

(0,0)

L0

OSC

L0
(1,0)

OSC

(0,1)
OSC

L0

(0,20)
OSC

(1,1)
OSC

L0

L0

OSC

i(k,l)OSC

L C

v(k,l)

L0
I. I NTRODUCTION
In this world, many kind of synchronization phenomenon
OSC
OSC (20,20)(20,0) OSC
can be observed. For example, we can observe the motions
Fig. 1. Circuit Model.
of many planets, schools of sardines, the synchronizationof fireflies, and so on. The synchronization phenomenon is named i(k,l) (see Fig. 1). The circuit equations of this circuita field which catches interest of many researchers. Then, model are normalized by Eq. (1), and the normalized circuit
the studies of synchronization phenomena are researched in equations are shown as Eqs. (2)–(6).
various fields[1],[2]. The synchronization phenomenon can be
Cg1
g1
i(k,l) = 3Lg
x,v=
observed on coupled oscillators system[3]. We are investigat(k,l)(k,l)
3g3 y(k,l) ,
3

(1)
√
ing the synchronization phenomena on coupled oscillators sysd
L
L
LCτ,
=
“
·
”,
α
=
,
ε
=
g
.
t
=
1
tem, and are developing some systems using synchronizationdτL0Cphenomena.
[Corner–top] (left:(a, b)=(0, 1). right:(a, b)=(19, 18).)
In our previous study, we observed synchronization phedx(0,a)
nomena on the ladder shape systems and lattice shape sys= y(0,a) ,(2)dτ
tems. On the ladder shape systems, the phase-waves, whichdy(0,a)= −x(0,a) + α(x(0,b) + x(1,a) − 2x(0,a) )are phenomena of propagating a phase difference betweendτ
1 3
adjacent oscillators, and the phase-inversion waves, which is+ ε(y(0,a) − y(0,a)).
3a phenomenon of propagating a phase state between adjacentoscillators, were discovered. The propagation and reflection [Corner–bottom] (left:(a, b)=(0, 1). right:(a, b)=(19, 18).)
mechanisms of the phase-waves and the phase-inversion waves
dx(19,a)= y(19,a) ,(3)
were clarified[4],[5]. On the lattice shape system, the phasedτ
waves and the phase-inversion waves were discovered[6],[7].
dy(19,a)= −x(19,a) + α(x(18,a) + x(19,b)dτPropagation and reflection mechanisms of the phase-waves
3
were clarified[6]. Moreover, we developed a prediction method− 2x(19,a) ) + ε(y(19,a) − 13 y(19,a)).
of time-series data using the lattice shape system. The timeseries data of Chua’s circuit was predicted by this system[8]. [Center] (0 < k < 19. 0 < l < 19.)dx(k,l)In this study, we clarify propagation mechanisms of the= y(k,l) ,(4)dτ
phase-inversion wave using instantaneous frequency of eachdy(k,l)oscillator and phase differences between adjacent oscillators= −x(k,l) + α(x(k+1,l) + x(k−1,l) + x(k,l+1) + x(k,l−1)dτ
on the lattice shape system. Furthermore, the air temperature3− 4x(k,l) ) + ε(y(k,l) − 13 y(k,l)).
data is predicted by the lattice shape system. The air temperature data is one of many natural data.
[Edge](top:(a, b)=(0, 1).bottom:(a, b)=(19, 18).both:0< l <19.)
II. C IRCUIT MODELdx(a,l)A lot of van der Pol oscillators are coupled by inductors= y(a,l) ,(5)dτL0 as a lattice(see Fig. 1). The number of column and thedy(a,l)number of row of this system are assumed as 20 respectively.
= −x(a,l) + α(x(a,l−1) + x(a,l+1) + x(b,l) − 3x(a,l) )dτ
We name each oscillator OSC(k,l). A voltage of each oscillator3+ ε(y(a,l) − 13 y(a,l)).
is named v(k,l) , and a current of inductor of each oscillator is978-1-4244-3896-9/09/$25.00 ©2009 IEEE

671

OSC(0,0)αβOSC(0,19)[Edge] (left:(a, b)=(0, 1). right:(a, b)=(19, 18). both:0<k<19.)
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) − 3x(k,a) )dτ
3+ ε(y(k,a) − 13 y(k,a)).

The α corresponds to the coupling parameter of each oscillator.
The ε corresponds to the nonlinearity of each oscillator. Thissystem is simulated by the fourth order Runge-Kutta methods
using Eqs. (2)-(6).

OSC(19,19)

OSC(19,0)

Fig. 2. The phase-inversion waves on 20x20 oscillators(α:an attractor of each
oscillator(current vs. voltage), β:a sum of voltages of adjacent oscillators(sumof voltage vs. time)).
τ(k, l)(a)V

III. P ROPAGATION MECHANISMSOSC(k, l)1.
τ(k,l) (a) − τ(k,l) (a − 1)

(7)

Three frequencies are observed in steady states. In this system, the synchronizations for vertical direction and horizontal
direction needs to be considered, because this system is 2
dimensional array. Therefore, three type synchronizations areobserved 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-phasesynchronization.
2. {(OSC(k, l)–OSC(k, l+1) and OSC(k, l)–OSC(k, l−1): inphase synchronization. OSC(k, l)–OSC(k + 1, l), and OSC(k,l)–OSC(k − 1, l): anti-phase synchronization.} or {OSC(k,l)–OSC(k, l + 1), and OSC(k, l)–OSC(k, l − 1): anti-phasesynchronization. OSC(k, l)–OSC(k + 1, l), and OSC(k, l)–OSC(k + 1, l): int-phase synchronization.}
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-phasesynchronization.
In this paper, we call the 1st type synchronization “in-and-inphase synchronization.” The 2nd type synchronization is called“in-and-anti-phase synchronization.” The 3rd type synchronization is called “anti-and-anti-phase synchronization.” Aneach instantaneous frequency of OSC(k, l) is obtained in eachsynchronization type. In the 1st situational synchronization,f(k,l) is fin−in . In the 2nd situational synchronization, f(k,l)
is fin−anti . In the 3rd situational synchronization, f(k,l) is
fanti−anti .
The phase difference is calculated as follows. A phasedifference between OSC(k, l) and OSC(k + 1, l) and a phasedifference between OSC(k, l) and OSC(k, l+1) are calculated.
The phase differences are assumed as Φ(k,l)(k+1,l) (a) andΦ(k,l)(k,l+1) (a) respectively(see Fig.3). The Φ(k,l)(k+1,l) (a)and Φ(k,l)(k,l+1) (a) are obtained by Eq.(8).

a

a+1

a+2

a+3

τ(k+1, l)(a)V

OSC(k+1, l)

a-1

a-2

a

a+1

a+2

a+3

Time

Fig. 3.

The detection method of frequencies and the phase differences.

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)× 180 [degree].
Φ(k,l)(k,l+1) (a) =τ(k,l) (a) − τ(k,l) (a − 1)Φ(k,l)(k+1,l) (a) =The phase-inversion waves are generated by the following
method. 1. Voltages and currents of all oscillators are set assame value, in other words all oscillators are set as in-phasesynchronization. 2. The signs of x(k,l) and y(k,l) of arbitrary
oscillators are inverted instantaneously.
A. Propagation mechanism I (propagation of phase-inversion
waves in in-and-in-phase synchronizations)
We can observe that a phase-inversion wave for a verticaldirection in each column propagate in in-and-in-phase synchronization. A mechanism is shown as Tab.I. A mechanismof a phase-inversion wave for a horizontal direction can be
explained same method. In Fig.4(a), the vertical axis is theinstantaneous frequency, and the horizontal axis is time. In
Fig.4(b), the vertical axis expresses the phase difference andthe horizontal axis expresses time.
fanti-ant i

Instantaneous Frequencyf(k,l) (a) =

a-2

Time

fin-ant i7f(7,1)

1

9f(6,1)
f(6,2)

fin-in3f(5,1)

5

Time[τ](a) Instantaneous frequency
180
6Phase Difference [degree]

A propagation of a phase-inversion waves can be observed(see Fig.2). The propagation mechanisms are madeclear using 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 namedf(k,l) (a) where “a” expresses the number of times of the peakvalue of the voltage. Time of a peak value of the voltage
of OSC(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).

a-1

8

Φ(7,1)(8,1)Φ(6,1)(7,1)Φ(5,1)(6,1)Φ(4,5)(5,1)Φ(6,1)(6,2)
024Time[τ](b) Phase difference

Fig. 4. Transitions of phase differences and frequencies by propagating aphase-inversion wave in in-and-in-phase synchronization.

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TABLE I
P ROPAGATION MECHANISM OF A PHASE - INVERSION WAVE IN
IN - AND - IN - PHASE SYNCHRONIZATION ( SEE F IG .4).

1

2
3

4
5
6
7
8
9

f anti-ant i

Instantaneous Frequency

MechanismΦ(8,0)(8,1) , Φ(8,1)(8,2) , Φ(7,0)(7,1) , Φ(7,1)(7,2) , Φ(6,0)(6,1) ,Φ(6,1)(6,2) , Φ(5,0)(5,1) and Φ(5,1)(5,2) are around 0 degrees
respectively. Then, a phase-inversion wave propagates for the vertical direction in in-and-in-phase synchronization. Now, the phaseineversion wave, which changes from in-phase synchronization toanti-phase synchronization, arrives at OSC(4,7).
f(7,1) starts to increase from fin−in to fin−anti , because
Φ(7,1)(8,1) is changed from 0 degree to 180 degrees by a phaseinversion wave.
Φ(6,1)(7,1) starts to increase from 0 degree to 180 degrees, because
f(7,1) changes from fin−in to fin−anti .
f(6,1) starts to increase from fin−in to fin−anti because
Φ(6,1)(7,1) increases toward 180 degrees. Instantaneous frequencyf(6,2) of an adjacent oscillator starts to increase at same timeby propagating a phase-inversion wave in the adjacent column.
Therefore, Φ(6,1)(6,2) is fixed as around 0 degrees. The phase statesof both sides are the same motion.
Φ(5,1)(6,1) starts to increase from 0 degree to 180 degrees, because
f(6,1) increases from fin−anti to fanti−anti .
f(5,1) starts to increase from fin−anti to fanti−anti , because
Φ(5,1)(6,1) increased from 0 degree to 180 degrees.
Φ(6,1)(7,1) becomes around 180 degrees, and a phase state between
OSC(6,1) and OSC(7,1) is fixed anti-phase synchronization.
f(6,1) arrives around fin−anti , because the phase state between
OSC(6,1) and OSC(7,1) is anti-phase synchronization.
Φ(5,1)(6,1) arrives around 180 degrees, and a phase state between
OSC(5,1) and OSC(6,1) is fixed anti-phase synchronization.
f(5,1) arrives around fanti−anti , because the phase state between
OSC(5,1) and OSC(6,1) is anti-phase synchronization.

7f (3,7)f(4,7)
f in-ant i

1f (3,8)9f(2,7)

5

3

f in-inTime[τ](a) Instantaneous frequency
180

Phase Difference [degree]

no.
0

8
Φ(3,7)(3,8)Φ(4,7)(5,7)

02

Φ(3,7)(4,7)

4Φ(2,7)(3,7)Φ(1,7)(2,7)

6Time[τ](b) Phase difference

Fig. 5. Transitions of phase differences and frequencies by propagating aphase-inversion wave in in-and-anti-phase synchronization.

TABLE IIP ROPAGATION M ECHANISM OF A PHASE - INVERSION WAVE IN
IN - AND - ANTI - PHASE SYNCHRONIZATION ( SEE F IG .5).
no.
0

B. Propagation mechanism II(propagating phase-inversion
waves in in-and-anti-phase synchronizations)
The phase states of adjacent oscillators of horizontal directions are anti-phase synchronization, and the phase statesof adjacent oscillators of vertical directions are in-phase synchronization. We can observe that a phase-inversion wave
for a vertical direction in each column propagates in in-andanti-phase synchronization. A mechanism is shown as Tab.II.
A mechanism of a phase-inversion wave for a horizontal
direction can be explained same method. In Fig.5(a), thevertical axis is the instantaneous frequency, and the horizontal
axis is time. In Fig.5(b), the vertical axis is the phase difference
and the horizontal axis is time.
IV. A PPLICATION FOR PREDICTION USING COUPLEDOSCILLATORS SYSTEM

We predict time-series data using the above system. Someparts of the system are changed something for the prediction
system. 1. The number of this system’s oscillators is fixed
nine. 2. 8 inductors(Lout(k,l) ) connecting to an arbitrary signal
generator are added to 8 oscillators of edges respectively,
because arbitrary time-series data is input from outside(seeFig.6). Therefore, this system has 17 inductors, 9 capacitors
and 9 nonlinear-negative-resistors. 3. These circuit elements
are adjusted independently.
A. Prediction method
The original time-series data is assumed as f (t). A prediction of the original time-series data is assumed as g(t). The
original time-series data of continuous data are divided by

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1

2
3

4
5
6
7
8
9

MechanismΦ(5,6)(5,7) , Φ(5,7)(5,8) , Φ(4,6)(4,7) , Φ(4,7)(4,8) , Φ(3,6)(3,7) ,Φ(3,7)(3,8) , Φ(2,6)(2,7) , Φ(2,7)(2,8) , Φ(1,6)(1,7) and Φ(1,7)(1,8)
are around 180 degrees respectively. Then, a phase-inversion wave
propagates for a vertical direction in in-and-anti-phase synchronization, that the phase states between adjacent oscillators ofrow direction are steadied as anti-phase synchronization and thephase states between adjacent oscillators of column direction are
steadied as in-phase synchronization. Now, the phase-ineversion
wave, which changes from in-phase synchronization to anti-phasesynchronization, arrives at OSC(4,7).
f4,7 starts to increase from fin−anti to fanti−anti , because
Φ(4,7)(5,7) ischanged from 0 degree to 180 degrees by a phaseinversion wave.
Φ(3,7)(4,7) starts to increase from 0 degree to 180 degrees, because
f(4,7) is changing toward fanti−anti .
f(3,7) starts to increase from fin−anti to fanti−anti , because
Φ(3,7)(4,7) is changed from 0 degree. Instantaneous frequencyf(3,8) of an adjacent oscillator starts to increase at same timeby propagating a phase-inversion wave in the adjacent column.
Therefore, Φ(3,7)(3,8) is fixed as around 180 degrees.
Φ(2,7)(3,7) starts to increase from 0 degree to 180 degrees, because
f(3,7) increases from fin−anti to fanti−anti .
f(2,7) starts to increase from fin−anti to fanti−anti , because
Φ(2,7)(3,7) increased from 0 degree to 180 degrees.
Φ(1,7)(2,7) starts to increase from 0 degree to 180 degrees, because
f(2,7) starts to increase from fin−anti to fanti−anti .
f(3,7) arrives around fanti−anti , because the phase state between
OSC(3,7) and OSC(4,7) is anti-phase synchronization.
Φ(2,7)(3,7) arrives around 180 degrees, and a phase state between
OSC(2,7) and OSC(3,7) is fixed anti-phase synchronization.
f(2,7) arrives around fanti−anti , because the phase state between
OSC(2,7) and OSC(3,7) is anti-phase synchronization.

Fig. 6.

Circuit model.
syntheis (prediction) data g(t)0

10T 10T

5T

Air Temperature [degree Celsius]

Fig. 8.

part of synthesis

15T 17T

20T(a) (b) (c)

(A)

(B)
0 original data f(t) 5T
20T

syntheis (prediction) data g(t)

25T

7T

10T

15T
part of prediction

Prediction data

28

Original data

3
April 1, 1981

Date

Fig. 9.

30T(d) (e) (f) (g) (h)10T

Error of prediction data and synthesis data.

20T

original data f(t)Fig. 7. Original data and synthesis data. (A), (B) and (C) are a continualgraph. 0[τ ] of original data is set at the left edge of (A). 0[tau] of synthesis
data is set at the left edge of (B). Prediction data can be observed in (C).

time period of arbitrary length T , and are predicted. We makea system which generates next data from previous data. The
next data and the previous data are used known-data. Then,
we make future data by using the system and the known-data.
The future data are unknown-data. The details of a procedureof our method, that the synthesis (or prediction) time-series
data g(t) are made from the original time-series data f (t), areshown as follows: 1. The original data from f (0) to f (10T )is added to our system, and data from g(0) to g(10T ) is
synthesized at the same time. The original data are shownin Fig. 7-(A). The synthesis data are shown in Fig. 7-(B). In
the Fig. 7, time axis of synthesis data differs from time axis oforiginal data. 2. The synthesized data g(t):(7T ≤ t ≤ 10T ) areadjusted to values close to f (t):(17T ≤ t ≤ 20T ) by changing
each parameter and eighteen initial values of oscillators. 3
The original time-series data from f (10T ) to f (20T ) is added
to the adjusted system, and time-series data from g(10T ) tog(20T ) is synthesized at the same time. The synthesis datag(t):(10T < t ≤ 20T ) are considered as prediction data.
However, to predict for long time is difficult, so a prediction
period is fixed from 10T to 15T . Then ,if the time-series datag(t):(10T < t ≤ 15T ) is close to f (t):(20T < t ≤ 25T ), we
can consider that g(t):(10T < t ≤ 15T ) predicts the originaltime-series data(see Fig. 7-(C)).
B. Experiment
In this paper, time-series data of the air temperature in
Hiroshima city from April 1981 to May 1985 is used as theoriginal time-series data. Then, f (t) is made by applying themoving average of every 31-day period, and subtracting an
average of a maximum and a minimum temperatures of thewhole period from each temperature.
In this study, domain (c) is thought to be more important
than (a) and domain (b)(see Fig. 7). Therefore, the average ofeach domain is weighted, and the average error of a synthesisdomain is calculated by the following equation.
ea × 1 + eb × 1.5 + ec × 2
synthesiserror =,(9)3
where ea , eb and ec express the average errors of domain(a), domain (b) and domain(c) respectively. If the total by

May 31, 1982

Prediction result.

the above method is around 0, good synthesis data is synthesized. We show a relationship between synthesis errors
and prediction errors(see Fig.8). The circuit parameters arechanged from local optimal solution. We fix some observation
condition as follows: 1. Only the coupling parameter ofOSC(1,1) is changed. 2. Averages of errors of prediction data
are calculated at the domains (d)–(h). We show the results oferrors of prediction data and synthesis data as Fig. 8. In thenatural complex time-series data, we can basically observe thatprediction error is largely minimum value when synthesis error
is minimum value.
The prediction result is shown in Fig. 9. We can observe thatthe prediction data exists around centerline of original data.
V. C ONCLUSION
We analyzed and clarified mechanism of propagation ofphase-inversion waves on the lattice shape system. The propagation mechanisms of phase-inversion waves, which exist
in in-and-in-phase synchronization and in-and-anti-phase synchronization, were explained by the instantaneous frequencyof each oscillator and the phase differences of adjacent oscillators. Furthermore, the air temperature data of Hiroshimacity was predicted by using the lattice shape system. We madeclear that our developed system can be used for prediction ofnatural complex data.
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.
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