2010 International Symposium on Nonlinear Theory and its Applications
NOLTA2010, Krakow, Poland, September 5-8, 2010

Propagation Mechanism of Phase-Inversion Wavein In-and-Anti-Phase Synchronization on 2D Lattice Oscillator
Seiko Kunihiro†, Hitoshi Aburatani† , Masayuki Yamauchi† and Yoshifumi Nishio‡†Department of Electronics and Computer Engineering,
Hiroshima Institute of Technology
2-1-1,Miyake, Saeki-ku, Hiroshima , Japan
Email: seiko.kunihiro@gmail.com‡Department of Electrical and Electronic Engineering,
Tokushima University
2-1, Minami-Josanjima, Tokushima, Japan(0,0)

Abstract—We analyze synchronization phenomena on
coupled oscillators systems as a ladder and a lattice. On thesystems, we observed phase-inversion waves, which are phenomena of changing phase states between two adjacent oscillators from in-phase synchronization to anti-phase synchronization or from anti-phase synchronization to in-phase synchronization in steady state. Some characteristics of phaseinversion waves are propagations, penetrations, reflections,and disappearances. In this paper, we discover the phaseinversion waves in in-and-anti-phase synchronization. We
clarify regions which the phase-inversion wave can be observed in in-and-anti-phase synchronization, and clarify amechanism of propagation of a phase-inversion wave in inand-anti-phase synchronization on the lattice system.
1. Introduction
A lot of synchronization phenomena can be observed innature world. For example, there are biological clocks,
schools of sardines, the synchronization of fireflies, and soon. Recently, synchronization phenomena are researched invarious fields[1]-[2].
In our previous study, we observed synchronization phenomena on coupled oscillators system. This system is madeby using van der Pol oscillators which are coupled by inductor as a lattice[3]. We predicted the time-series data byusing this system including nine oscillators[4]. We observed
phase-inversion waves on this system including over 25 oscillators. We analyzed a mechanism of disappearance between two phase-inversion waves. Further, we analyzed amechanism of reflection when two phase-inversion waves
arrive at a corner at same time. However, these phaseinversion waves are observed in double-in-phase synchronization which all oscillators synchronize to in-phase for a
vertical direction and a horizontal direction. In other hand,on ladder system, the phase-inversion waves are observedin in-and-anti-phase synchronization[5]. In-and-anti-phasesynchronization is in-phase and anti-phase synchronizations
exist alternately.
In this study, we observe the phase-inversion waves in inand-anti-phase synchronization. We clarify regions which
the phase-inversion wave can be observed in in-and-antiphase synchronization when N equals 9, and clarify a mechanism of propagation of a phase-inversion wave in in-and-

L0

OSC

(0,1)
OSC

L0

L0

(0,N)
OSC

L0
(1,1)
(1,0)

OSC

OSC

OSC

i(k,l)

L0

OSC

L C

v(k,l)

L0
(N,0)

OSC

OSC

OSC

(N,N)

Figure 1: Circuit model.
anti-phase synchronization using instantaneous frequency ofeach oscillator and phase diﬀerences between adjacent oscillators on the lattice system.
2. Circuit model
The van der Pol oscillators are coupled by inductors L0 as
a lattice(see Fig. 1). The number of column and row of thissystem are assumed as “N + 1” respectively. We name each
oscillator OSC(k,l). A voltage of each oscillator is namedv(k,l) , and a current of inductor of each oscillator is namedi(k,l) (see Fig. 1). The circuit equations of this 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
t = LCτ, dτ= “ · ”, α = LL0 , ε = g1 CL .
[Corner–top] (left:(a, b)=(0, 1). right:(a, b)=(N, N − 1).)dx(0,a)dτ = y(0,a) ,dy(0,a)dτ = −x(0,a) + α(x(0,b) + x(1,a) − 2x(0,a) )(2)

1+ ε(y(0,a) − y3(0,a) ).
3[Corner–bottom] (left:(a, b)=(0, 1). right:(a, b)=(N, N − 1).)dx(N,a)dτ = y(N,a) ,dy(N,a)dτ = −x(N,a) + α(x(N−1,a) + x(N,b) − 2x(N,a) )(3)+ ε(y(N,a) − 13 y3(N,a) ).
[Center] (0 < k < N. 0 < l < N.)
dx(k,l)dτ = y(k,l) ,- 15 -(4)dy(k,l)dτ = −x(k,l) + α(x(k+1,l) + x(k−1,l) + x(k,l+1) + x(k,l−1)− 4x(k,l) ) + ε(y(k,l) − 13 y3(k,l) ).

[Edge](top:(a, b)=(0, 1).bottom:(a, b)=(N, N − 1).both:0< l <N.)
dx(a,l)dτ = y(a,l) ,dy(a,l)dτ = −x(a,l) + α(x(a,l−1) + x(a,l+1) + x(b,l) − 3x(a,l) )OSC(0,0)A

B

OSC(0,8)y(0,10)+y(1,10)y(1,10)+y(2,10)y(2,10)+y(3,10)(5)y(3,10)+y(4,10)y(4,10)+y(5,10)y(5,10)+y(6,10)+ ε(y(a,l) − 13 y3(a,l) ).
(left:(a, b)=(0, 1). right:(a, b)=(N, N − 1). both:0<k<N.)
dx(k,a)dτ = y(k,a) ,dy(k,a)dτ = −x(k,a) + α(x(k−1,a) + x(k+1,a) + x(k,b) − 3x(k,a) )(6)y(6,10)+y(7,10)y(7,10)+y(8,10)
OSC(8,0)OSC(8,8)Figure 2: Pattern A - Phase-inversion waves on 9x9
oscillators(α=0.05 and ε=0.15).
y(0,10)+y(1,10)y(1,10)+y(2,10)+ ε(y(k,a) − 13 y3(k,a) ).

3. In-and-anti-phase synchronization
In our circuit, an oscillator, which is not an oscillator
on the edge, has four adjacent oscillators. When phasestates between the oscillator and two of four oscillators are
anti-phase synchronization, phase states between the oscillator and other two oscillators are in-phase synchronization. Oscillators on the edges stay in anti-phase synchronization in in-and-anti-phase synchronization. These phasestates are called “in-and-anti-phase synchronization.” Thephase-inversion waves in in-and-anti-phase synchronization
are classified into two patterns. Pattern A can be observed if
N is an odd number. Odd number’s phase-inversion waves
propagate in vertical direction and horizontal direction respectively. Pattern B can be observed if N is an even number.
Even number’s phase-inversion waves propagate in verticaldirection and horizontal direction respectively. Simulation
results of pattern A and B show in Figs. 2 and 3 respectively.
Figure 4 shows regions which the phase-inversion wave canbe observed in in-and-anti-phase synchronization when N
equals 9. The coupling parameter α and nonlinearity ε arechanged from 0.050 to 1.0, every 0.050. The phase-inversion
wave in in-and-anti-phase synchronization can be observedin region(i)(see Figs. 2 and 4). The complex phenomena on
in-and-anti-phase synchronization can be observed in region(ii)(see Figs. 4 and 5).
We can observe some characteristics of phase-inversion
waves in in-and-anti-phase synchronization. These characteristics are a propagation, a penetration, a reflection
at an edge, and a reflection between two phase-inversion
waves(see Figs. 2 and 3). These characteristics are shownin Table 1.
4. Mechanism
We analyze a mechanism of propagation of a phaseinversion wave. The mechanism is made clear by using instantaneous frequency of each oscillator and phase diﬀerences between adjacent oscillators. Figure 6 shows the signs- 16 -

y(3,10)+y(4,10)y(4,10)+y(5,10)y(5,10)+y(6,10)y(6,10)+y(7,10)y(7,10)+y(8,10)y(8,10)+y(9,10)

Figure 3: Pattern B - Phase-inversion waves on 10x10
oscillators(α=0.05 and ε=0.15).
of the initial values of the voltages and currents of each oscillator. The coupling parameter α is fixed as 0.05, and nonlinearity ε is fixed as 0.15. An equation of the instantaneous
frequency of OSC(k, l) is obtained as follows(see Eq. (7)).
The instantaneous frequency is named f(k,l) (a) where “a” expresses the number of times of the peak value of the voltage. Time of a-th peak value of the voltage of OSC(k, l)
is assumed as τ(k,l) (a)(see Fig. 7). Similarly, τ(k+1,l) (a) andτ(k,l+1) (a) are decided.
1.
(7)
τ(k,l) (a) − τ(k,l) (a − 1)
Three frequencies are observed in this system. To considerof the synchronizations for the vertical direction and for thehorizontal direction are needed, because this system is 2 dimensional array. The in-phase synchronization and the antiphase synchronization exist. Therefore, three types of 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): the in-phase synchronization.
2. {(OSC(k, l)–OSC(k, l − 1), and OSC(k, l)–OSC(k, l + 1)
are a same phase synchronization state, and OSC(k, l)–OSC(k − 1, l), and OSC(k, l)–OSC(k + 1, l) are anotherphase synchronization state.} or {OSC(k, l)–OSC(k − 1,f(k,l) (a) =

1

(ii)

ε

The α corresponds to a coupling parameter. The ε corresponds to a nonlinearity of each oscillator. This system
is simulated by the fourth order Runge-Kutta method andEqs. (2)-(6). The phase-inversion waves are shown in Fig. 2.
The Fig. 2–A expresses an attractor of each oscillator(current
vs. voltage). The Fig. 2–B expresses itinerancy of phase difference by which sum of voltages of adjacent oscillators isshown along the time(sum of voltage vs. time).

y(2,10)+y(3,10)

0.05
0.05

α

1

Figure 4: Region of in-and-anti-phase synchronization on
9x9 oscillators

Table 1: Characteristics of the phase-inversion waves on in-and-anti-phase synchronization.
Names of characteristics

Phenomena
The phase-inversion waves propagate for vertical direction or horizontal direction. The vertical phase-inversion
waves independently move from the horizontal phase-inversion waves.

Propagations

Two phase-inversion waves arrive at an oscillator from vertical direction and horizontal direction, and each phase-

Penetrations

inversion wave penetrates each other.
When a phase-inversion wave arrives at an edge, the phase-inversion wave reflects and propagates to where they

Reflections at an edge

came from. Sometime this phenomenon is happened with penetration.

Reflections between twophase-inversion waves

When two phase-inversion waves coming from the opposite directions arrive to two adjacent oscillator at same time,
the phase-inversion waves reflect and propagate to where they came from.
τ(k, l)(a)V

OSC(k, l)

a-1

a

a+1

Time

τ(k+1, l)(a)V

OSC(k+1, l)
a-1

a

a+1

Time

Figure 5: An example of complex phenomena inregion(ii)(α=0.05 and ε=0.85).
0

2

3

4

5

6

7

fanti-anti

8+ + − − + + − − ++ + − − + + − − +− − + + − − + + −− − + + − − + + −+ + − − + + − − ++ + − − + + − − +− − + + − − + + −− − + + − − + + −+ + − − + + − − +Instantaneous Frequency

0
1
2
3
4
5
6
7
8

1

Figure 7: The detection method of frequencies and the phase
diﬀerences.

τ(k,l) (a) − τ(k+1,l) (a)× 360 [degree]τ(k,l) (a) − τ(k,l) (a − 1)τ(k,l) (a) − τ(k,l+1) (a)× 360 [degree].
Φ(k,l)(k,l+1) (a) =τ(k,l) (a) − τ(k,l) (a − 1)Φ(k,l)(k+1,l) (a) =(8)

4.1. Propagation mechanism
We can observe a phenomenon that a phase-inversion
wave to a vertical direction in each column propagate in in-- 17 -

f in-antif(9,3)
94

f(8,3)

118

f in-inFigure 6: Sign of initial value of each oscillator of in-andanti-phase synchronization.

Time[ ](a) Instantaneous frequency
36010(8,3)(9,3)(9,2)(9,3)

3Phase Difference [degree]l), and OSC(k, l)–OSC(k, l − 1): are a same phase synchronization state, and OSC(k, l)–OSC(k + 1, l), andOSC(k, l)–OSC(k, l + 1): are another phase synchronization state.} or {OSC(k, l)–OSC(k − 1, l), and OSC(k,l)–OSC(k, l+1): are a same phase synchronization state,
and OSC(k, l)–OSC(k, l − 1), and OSC(k, l)–OSC(k + 1,l): are another phase synchronization state.}
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): the anti-phase synchronization.
An instantaneous frequency f(k,l) of OSC(k, l) is obtained ineach synchronization-type. The 1st situational synchronization frequency is called fin−in . The 2nd situational synchronization frequency is called fin−anti . The 3rd situational synchronization frequency is called 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 obtained. 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)(see Fig. 7).

5
2

180
6

0
(7,3)(8,3)

12

-180
-360
-540

1

(9,3)(9,4)(9,3)(10,3)
7

Time[ ](b) Phase difference

Figure 8: Transitions of phase diﬀerence and frequencies by
propagation of a phase-inversion wave on in-and-anti-phasesynchronization.
and-anti-phase synchronization. We fix the N as 19. Propagation mechanism is shown in Table. 2(see Fig. 8). In Fig.
8(a), the vertical axis is instantaneous frequency, and horizontal axis is time. In Fig. 8(b), the vertical axis is the phase
diﬀerence, and the horizontal axis is time.
4.2. Comparison between a propagation in double inphase synchronization and a propagation in in-andanti-phase synchronization.
The frequency’s itinerancies of propagation of the phaseinversion wave in double in-phase synchronization are Fig.
9. The frequencies are changed from fin−in to fanti−anti .
5. Conclusion
We discovered the phase-inversion waves in in-and-antiphase synchronization. We clarified regions which the phase-

Table 2: Propagation mechanism of a phase-inversion wave(see Fig. 8).
no.

Mechanism

0

At this time, Φ(9,3)(9,4) and Φ(8,3)(8,4) are fixed the in-phase synchronization. Φ(9,2)(9,3) and Φ(8,2)(8,3) are fixed the anti-phase synchronization.
In vertical direction, the phase-inversion wave, which changes synchronized state, arrives at the number of row is 10 from the number of row is

1

A phase diﬀerence Φ(9,3)(10,3) starts to change from the in-phase synchronization toward the anti-phase synchronization by a phase-inversion
wave.

2

A instantaneous frequency f(9,3) starts to increase from fin−anti toward fanti−anti , because Φ(9,3)(10,3) starts to change from the in-phase synchro-18.

nization toward the anti-phase synchronization and Φ(9,2)(9,3) and Φ(8,3)(9,3) are anti-phase synchronization and Φ(9,3)(9,4) is in-phase synchronization.
Φ(8,3)(9,3) starts to change from the anti-phase synchronization toward the in-phase synchronization by f(9,3) .

3

f(8,3) starts to decrease from fin−anti toward fin−in , because Φ(8,3)(9,3) starts to change from the anti-phase synchronization toward the in-phase4synchronization and Φ(8,3)(8,4) and Φ(7,3)(8,3) are the in-phase synchronization and Φ(8,2)(8,3) is the anti-phase synchronization.
f(9,3) doesn’t arrive at fanti−anti and f(9,3) starts to decrease toward fin−anti again, because Φ(9,3)(10,3) starts to change from the in-phase syn-5chronization toward the anti-phase synchronization and Φ(8,3)(9,3) starts to change from the anti-phase synchronization toward the in-phasesynchronization and Φ(9,3)(9,4) is the in-phase synchronization and Φ(9,2)(9,3) is the anti-phase synchronization.
6

Φ(7,3)(8,3) starts to change from the in-phase synchronization toward the anti-phase synchronization by f(8,3) .

7

Φ(9,3)(10,3) arrives at the anti-phase synchronization and becomes fix.
f(8,3) doesn’t arrive at fin−in and f(8,3) starts to increase toward fin−anti again, because Φ(8,3)(9,3) starts to change from the anti-phase synchroniza-

8

tion toward the in-phase synchronization and Φ(7,3)(8,3) starts to change from the in-phase synchronization toward the anti-phase synchronizationand Φ(8,3)(8,4) is the in-phase synchronization and Φ(8,2)(8,3) is the anti-phase synchronization.
f(9,3) arrives at fin−anti again, because Φ(9,3)(10,3) arrives at the anti-phase synchronization and Φ(8,3)(9,3) starts to change from the anti-phase9synchronization toward the in-phase synchronization and Φ(9,3)(9,4) is the in-phase synchronization and Φ(9,2)(9,3) is the anti-phase synchronization.
10Φ(8,3)(9,3) arrives at in-phase synchronization and becomes fix, because f(9,3) arrives at fin−anti and becomes fix.

11

f(8,3) arrives at fin−anti again, because Φ(8,3)(9,3) arrives at the in-phase synchronization and Φ(7,3)(8,3) starts to change from the in-phase synchronization toward the anti-phase synchronization and Φ(8,3)(8,4) is the in-phase synchronization and Φ(8,2)(8,3) is the anti-phase synchronization.

12

Φ(7,3)(8,3) arrives at anti-phase synchronization and becomes fix, because f(8,3) arrives at fin−anti and becomes fix.

The phase-inversion wave propagates by this mechanism.
Horizontal phase-inversion wave
has not arrive yet.

Vertical phase-inversion wave
already past.

Instantaneous Frequency

fanti-antif(6,2)
f in-antif(7,1)

f(5,1)
f(6,1)

f in-inTime[ ](a) Instantaneous frequency

Figure 9: Transitions of frequencies by propagation of aphase-inversion wave on in-and-in-phase synchronization.
inversion wave can be observed in in-and-anti-phase synchronization when N equals 9, and clarified a mechanism of
propagation of a phase-inversion wave in in-and-anti-phasesynchronization by using instantaneous frequency of each
oscillator and phase diﬀerences between adjacent oscillators on the lattice system, and compared between a propagation in double in-phase synchronization and a propagation
in in-and-anti-phase synchronization. The frequencies ofthe phase-inversion wave in double in-phase synchronization
are changed from fin−in to fanti−anti . However, the frequencies of the phase-inversion wave in in-and-anti-phase synchronization are changed around fin−anti , and can not change
to fin−in and fanti−anti . We observed some characteristicsof phase-inversion waves on in-and-anti-phase synchronization. These characteristics are a propagation, a penetration,- 18 -

a reflection at an edge, and a reflection between two phaseinversion waves.
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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