18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

Penetration and Reflection Mechanisms of
Phase-Inversion Waves in Lattice Oscillators
Hitoshi ABURATANI, Suguru YAMANE
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—We observe and analyze particular synchronization
phenomena on a lattice of coupled oscillators. Wave-motion
phenomena which propagate phase differences can be observedon the system in steady states, when the system has nonlinearity.
We call this wave-motion phenomena “phase-inversion waves.”
The phase-inversion waves have some behaviors, which arepropagation, reflection, extinction, and penetration. In this paper,a penetration mechanism between a vertical phase-inversion wave
and a horizontal phase-inversion wave is made clear. Furthermore, a reflection mechanism between two phase-inversion waves
is made clear.

name each oscillator OSC(k,l). A voltage of each oscillator isnamed v(k,l) , and a current of inductor of each oscillator isnamed i(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
i(k,l) = 3Lg
x(k,l) , v(k,l) = 3gy(k,l) ,
3
3

(1)
√
d
L
t = LCτ, dτ= “ · ”, α = LL0 , ε = g1 C.

[Corner–top] (left:(a, b)=(0, 1). right:(a, b)=(N, N − 1).)

I. I NTRODUCTION

dx(0,a)= y(0,a) ,dτdy(0,a)= −x(0,a) + α(x(0,b) + x(1,a) − 2x(0,a) )dτ

II. C IRCUIT MODELdx(k,a)= y(k,a) ,(6)dτdy(k,a)= −x(k,a) + α(x(k−1,a) + x(k+1,a) + x(k,b) − 3x(k,a) )dτ

(2)In this world, there are synchronization phenomena in every
where[1]. The synchronization phenomena are existing among
1 3).
+ ε(y(0,a) − y(0,a)
atoms, sea waves, planets and so on. There are synchronization3
phenomena also among humans as rumors, trends, and soon. Therefore, the synchronization phenomenon is researched [Corner–bottom] (left:(a, b)=(0, 1). right:(a, b)=(N, N − 1).)dx(M,a)in various fields. For example, there is research of synchro= y(M,a) ,(3)dτnization phenomena of a lot of fireflies[2].We investigatedy(M,a)= −x(M,a) + α(x(M−1,a) + x(M,b)
dτand analyze the synchronization phenomena on the coupled3
oscillator systems[3]. In our previous study, we discovered− 2x(M,a) ) + ε(y(M,a) − 13 y(M,a)).
the wave-motion which propagates phase difference, which isaround 180 degrees, between adjacent oscillato rs on coupled [Center] (0 < k < M . 0 < l < N .)dx(k,l)oscillators system as a ladder and as a lattice. We call= y(k,l) ,(4)dτ
the wave-motion “phase-inversion wave.” The phase-inversiondy(k,l)= −x(k,l) + α(x(k+1,l) + x(k−1,l) + x(k,l+1) + x(k,l−1)
waves can be observed in steady states[4]. There are thedτ3phase-inversion wave propagating to vertical direction and the− 4x(k,l) ) + ε(y(k,l) − 13 y(k,l)).
phase-inversion wave propagating to horizontal direction on[Edge]
the lattice coupled oscillators system.
(top:(a, b)=(0, 1).bottom:(a, b)=(M, M−
1).both:0<
In this study, attracting forces to in-phase or anti-phasesynchronization are investigated on one parameter-set where l <N .)dx(a,l)the phase-inversion waves can be observed. A penetration= y(a,l) ,(5)dτmechanism between the vertical phase-inversion wave anddy(a,l)= −x(a,l) + α(x(a,l−1) + x(a,l+1) + x(b,l) − 3x(a,l) )the horizontal phase-inversion wave and reflection mechanismdτ
between two phase-inversion waves are made clear by instanta3).
+ ε(y(a,l) − 13 y(a,l)
neous frequencies of each oscillator, phase differences between(left:(a, b)=(0, 1). right:(a, b)=(N, N − 1). both:0<k<M .)adjacent oscillators, and the attracting forces.
A lot of van der Pol oscillators are coupled by inductors L0
as a lattice(see Fig. 1). The number of column and the numberof row of this system are assumed as “M +1” and “N +1.” We
26-28 May 2010, Dresden, Germany

154

3
).
+ ε(y(k,a) − 13 y(k,a)

L0

OSC

(0,1)

L0

L0

OSC

(0,N)

anti-phase

(0,0)

OSC

L0
(1,1)
OSC

OSC

OSC

L0

OSC

L C

OSC

OSC

OSC

3
1

v(k,l)

L0
(M,0)

4
2

i(k,l)

in-phase(1,0)[degree]18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

(M,N)

1
2
0

Fig. 1.

Circuit Model.

20

40

60

A B

Fig. 3.

Fig. 4.

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)).

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). The phase-inversion waves are shown in
Fig. 2. Figure 2–A expresses attractor of each oscillators.
Figure 2–B expresses itinerancy of phase difference sum of
voltages of adjacent oscillators is shown along the time.
III. ATTRACTING FORCE
Attracting forces to in-phase or anti-phase synchronization are investigated on one parameter-set where the phaseinversion waves can be observed(see Fig.3). Attracting forces
are observed as follows:
1) The phase differences between all adjacent oscillators
are fixed as arbitrary value in the lattice system.
2) The phase difference between OSC(1,4) and OSC(1,5)
after one period from initial value is analyzed along theinitial phase difference is changed.
A vertical axis of Fig.3 expresses a variation of phase difference per one cycle. An upper direction shows attracting force
to in-phase. A downward direction shows attracting force toanti-phase. A horizontal axis shows initial phase differences.
Therefore, the length of line shows a attracting forces at eachphase difference. Attracting force to in-phase is the strongest
in 40 degrees. Attracting force to anti-phase is the strongest
in 130 degrees.
IV. B EHAVIOR OF PHASE - INVERSION WAVES
We can observed some behaviors of phase-inversion waves
on above systems. These behaviors are a propagation, a reflection at an edge, a reflection between phase-inversion waves,distinction and a penetration. Moreover, these behaviors can
be classified by frequencies, because three frequencies areobserved in steady states. The synchronizations for vertical
26-28 May 2010, Dresden, Germany

80 100 120 140 160 180[degree]

Initial phase differences

Attracting forces.

The detection method of frequencies and the phase differences.

direction and for horizontal 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 synchronization.
2) {(OSC(k, l)–OSC(k, l+1) and OSC(k, l)–OSC(k, l−1):in-phase synchronization. OSC(k, l)–OSC(k + 1, l), andOSC(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-phase synchronization. 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-phase synchronization.
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 f(k,l) of OSC(k, l) is obtained ineach synchronization 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 .
These behaviors are shown in Table I. There is a disappearance by collision of two phase-inversion waves other thanthese phenomena.
V. M ECHANISM
We analyze about the penetration and reflection of the twophase-inversion waves.
The phase-inversion wave shows in Fig.2. The mechanisms
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 isfixed as ε = 0.05. An equation of the instantaneous frequency

155

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)
TABLE IC HARACTERISTICS OF THE PHASE - INVERSION WAVES .
Itinerancies of instanta-

Names of behaviors

Phenomena

neous frequencies

Propagations

fin−in ⇔ fin−anti ,&fin−anti ⇔ fanti−anti

Penetrations

fin−in ⇔ fanti−antiReflections at an edge

fin−in ⇔ fin−anti ,&fin−anti ⇔ fanti−anti

When an phase-inversion wave arrives at an edge, the phase-inversion wave reflects
and propagates to where they came from. Sometime this phenomenon is happened with

Reflections between two

fin−in ⇔ fin−anti ,&fin−anti ⇔ fanti−anti

When two phase-inversion waves coming from the opposite directions arrive to twophase-inversion waves

The phase-inversion waves propagate to vertical direction or horizontal direction.
The vertical phase-inversion waves move from the horizontal phase-inversion waves
independently.
Two phase-inversion waves arrives at an oscillator from vertical direction and horizontal
direction, and each phase-inversion wave penetrates each other.

penetration.
adjacent oscillator at same time, the phase-inversion waves reflect and propagate towhere they came from.

Instantaneous Frequency

Instantaneous Frequency

fanti-antif(5,5)
f(4,6)
andf(6,5)

f(6,4)

andfin-antif(5,6)
f(5,7)
andf(5,4)

f(7,5)

fin-inandf(4,5)

1

fanti-antif(4,5)
andfin-antif(5,5)
4f(3,5)

andandf(7,5)

f(6,5)

fin-inTime[τ](a)Instantaneous frequency

Phase Difference [degree]

Time[τ](a)Instantaneous frequency

Phase Difference [degree]3f(2,5)

3180
Φ(5,4)(6,4)
160
and
140
Φ(4,5)(4,6)
120
100 Φ(5,6)(5,7)
80
and60 Φ(6,5)(6,7)
40
20
2
0
-20

1

Φ(5,5)(6,5)
and

Φ(5,5)(5,6)Φ(4,5)(5,5)
and

Φ(5,4)(5,5)
4

900
720
540
360
180

Φ(3,5)(4,5)

52Φ(2,5)(3,5)

0

-180
-360
-540
-720
-900

Φ(4,5)(5,5)

25Φ(5,5)(6,5)Φ(6,5)(7,5)

Time[τ](b)Phase difference

Time[τ](b)Phase difference

Fig. 5. Transitions of phase difference and frequencies by penetration aphase-inversion wave.

Fig. 6. Transitions of phase difference and frequencies by reflection between
two phase-inversion wave.

of OSC(k, l) is calculated as follows. The instantaneous
frequency is named f(k,l) (a) where “a” expresses the numberof times of the peak value of the voltage. Time of a peakvalue of the voltage of OSC(k, l) is assumed as τ(k,l) (a)(seeFig.4). Similarly, τ(k+1,l) (a) and τ(k,l+1) (a) are decided. Thef(k,l) (a) is obtained by Eq.(7).

A. Penetration Mechanism

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

(7)

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.4). 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)× 360 [degree]τ(k,l) (a) − τ(k,l) (a − 1)τ(k,l) (a) − τ(k,l+1) (a)Φ(k,l)(k,l+1) (a) =× 360 [degree].
τ(k,l) (a) − τ(k,l) (a − 1)

We can observe the phenomenon of penetration. A mechanism of penetration is shown in Tab. II and Fig. 5. The verticalaxis of Fig. 5(a) is the instantaneous frequency, and horizontal
direction is time. The vertical axis of Fig. 5(b) expresses thephase differences, and horizontal direction expresses time.
B. Reflection mechanisms between two phase-inversion waves
We can observe the reflection phenomenon between twophase-inversion waves which are arrives at two adjacent oscillators at same time from opposite directions respectively. We
shows mechanism in Table III. In Fig.6(a), the vertical axis is
the instantaneous frequencies, and the horizontal axis is time
In Fig.6(b), the vertical axis is the phase differences, and thehorizontal axis is the time.
VI. C ONCLUSION

Φ(k,l)(k+1,l) (a) =26-28 May 2010, Dresden, Germany(8)

An attracting forces to in-phase and to anti-phase were madeclear by all adjacent oscillators being fixed as arbitrary phase156

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)
TABLE IIP ENETRATION MECHANISM ( SEE F IG .5).
no.

Vertical direction

Horizontal direction

0

Now, phase-inversion waves, which changes phase states from in-phasesynchronization to anti-phase synchronization, are arrived at oscillators

Now, phase-inversion waves, which changes phase states from in-phasesynchronization to anti-phase synchronization, are arrived at oscillatorsof 7th row line from oscillators of 8th row line.

of 7th column line from oscillators of 8th column line.

Φ(6,5)(7,5) is increasing toward 180 degrees. f(6,5) slowly starts toΦ(5,6)(5,7) is increasing toward 180 degrees. f(5,6) slowly starts toincrease from fin−in to fin−anti , after Φ(6,5)(7,5) start to increase,because an attracting force to in-phase is weak around 0 degree(seeincrease from fin−in to fin−anti , after Φ(5,6)(5,7) start to increase,because an attracting force to in-phase is weak around 0 degree(seeFig.3). f(6,4) also starts to increase from fin−in , because a phase-

Fig.3). f(4,6) also starts to increase from fin−in , because a phase-

inversion wave of adjacent vertical direction arrives at 6th row line at
the same time. f(6,4) changes toward fin−anti , because horizontal

inversion wave of adjacent vertical direction arrives at 6th column line
at the same time. f(4,6) changes toward fin−anti , because horizontal

1

wave does not arrives at OSC(6,4) yet.

wave does not arrives at OSC(4,6) yet.

2Φ(5,5)(6,5) and Φ(5,4)(6,4) start to increase toward 180 degrees,
because f(6,5) and f(6,4) increase toward fin−anti .

Φ(5,5)(5,6) and Φ(4,5)(4,6) start to increase toward 180 degrees,
because f(5,6) and f(4,6) increase toward fin−anti .

3

A horizontal phase-inversion wave and a vertical phase-inversion wave arrives at OSC(5,5). f(5,5) starts to increase to fanti−anti byhorizontal phase-inversion wave and vertical phase-inversion wave rapidly. f(4,5) and f(5,4) start to increase from fin−in to fanti−anti ,because Φ(5,4)(6,4) and Φ(4,5)(4,6) start to increase toward 180 degrees.

4

Therefore, Φ(4,5)(5,5) starts to increase toward 180 degrees because
f(5,5) increases toward fanti−anti . Because an attracting force to

Therefore, Φ(5,4)(5,5) starts to increase toward 180 degrees because
f(5,5) increases toward fanti−anti . Because an attracting force toanti-phase is weak around 180 degrees, Φ(4,5)(5,5) arrives at 180

anti-phase is weak around 180 degrees, Φ(5,4)(5,5) arrives at 180

degrees slowly(see Fig.3).

degrees slowly(see Fig.3).

The phase-inversion waves penetrate each other by above mechanism.
TABLE IIIR EFLECTION MECHANISM BETWEEN TWO PHASE - INVERSION WAVES ( SEE F IG .6).
no.

upside0

Now the phase states of horizontal direction are fixed in in-phase synchronization. The phase states of vertical directions around OSC(5,5)
are anti-phase synchronization The phase-inversion wave changing from anti-phase to in-phase synchronization arrives on 3rd row and 6th
row line from 0th row line and 9th row line respectively.

down sidef(3,5) starts to decrease from fin−anti to fin−in , because

f(6,5) starts to decrease from fin−anti to fin−in , because

Φ(2,5)(3,5) started to change from anti-phase to in-phase. f(3,5)
slowly changes, because an attracting force is weak force around anti-Φ(6,5)(7,5) started to change from anti-phase to in-phase. f(6,5)
slowly changes, because an attracting force is weak force around anti-

phase synchronization(see Fig.3).

phase synchronization(see Fig.3).

2

Φ(3,5)(4,5) starts to changing from anti-phase to in-phase, because
f(3,5) started to decrease from fin−anti to fin−in .

Φ(5,5)(6,5) starts to change from anti-phase to in-phase, because
f(5,5) started to decrease from fin−anti to fin−in .

3f(4,5) starts to decrease from fin−anti to fin−in slowly, because

f(5,5) starts to decrease from fin−anti to fin−in slowly, because

Φ(3,5)(4,5) started to change from anti-phase to in-phase.

Φ(5,5)(6,5) started to change from anti-phase to in-phase.

14

Φ(4,5)(5,5) are fixed anti-phase, because the frequencies of OSC(4,5) and OSC(5,5) are same itinerancy. Therefore, f(4,5) and f(5,5) stop
between fin−anti and fin−in , and f(4,5) and f(5,5) start to increase to fin−anti again.

5Φ(3,5)(4,5) can not stop at in-phase synchronization, because reflection phase-inversion wave arrives in a moment again.

Φ(4,5)(5,5) can not stop at in-phase synchronization, because reflection phase-inversion wave arrives in a moment again.

Two phase-inversion waves reflect by above mechanism.

difference.
A penetration mechanism between the vertical phaseinversion wave and horizontal phase-inversion wave was madeclear by instantaneous frequencies of each oscillator, phasedifferences between adjacent oscillators and the attracting
forces. Furthermore, a reflection mechanism between twophase-inversion waves was made clear by instantaneous frequencies of each oscillator, phase differences between adjacentoscillators and the attracting forces.
ACKNOWLEDGEMENTS

for the Promotion of Science.
R EFERENCES[1] B. Schechter, “How the brain gets rhythm,” Science., vol. 274, no. 5286,
pp. 339-340, Oct. 1996.
[2] N. Ohba, “Flash Communication Systems of Japanese Fireflies,” Integrative and Comparative Biology,vol.44, pp.225-233, June 2004.
[3] T. Endo and S. Mori, “Mode Analysis of Two-Dimensional Low-Pass
Multimode Oscillator,” IEEE Trans. Circuits and Syst., vol. 23, no. 9, pp.
517-530, Sept. 1976.
[4] M. Yamauchi, M. Okuda, Y. Nishio and A. Ushida, “Analysis of PhaseInversion Waves in Coupled Oscillators Synchronizing at In-and-AntiPhase,” IEICE Trans. Fundamentals, vol.E86-A, no.7, pp.1799-1806, July
2003.

This research is supported by the Grants-in-Aid for Young
Scientific Research (B) (No. 19760270) from the Japan Society
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