2010 International Workshop on Nonlinear Circuits,
Communication and Signal Processing
NCSP'10, Waikiki, Hawaii, March 3-5, 2010

Me hanisms of Disappearan e between Two Phase-Inversion Waves and Ree tion of Two
Phase-Inversion Waves at a Corner on 2D Latti e Os illators

y

y

y

Seiko Kunihiro , Hitoshi Aburatani , Suguru Yamane , Masayuki Yamau hi

yDepartment of Ele troni s and Computer Engineering,
Hiroshima Institute of Te hnology
2-1-1, Miyake, Saeki-ku, Hiroshima , Japan
Email: seiko.kunihirogmail. omyand Yoshifumi NishiozzIntelligent Networks and Computer S ien es,
Department of Ele tri al and Ele troni Engineering,
Tokushima University[Cornertop℄ (left:(a; b)=(0; 1). right:(a; b)=(N; N 1).)
Abstra t
We observe and analyze syn hronization phenomena on oudx a(2)d = y(0;a) ;pled os illators system. We dis overed the phase-inversion
dy awave, whi h phenomena of hanging phase states betweend = x(0;a) + ( x(0;b) + x(1;a) 2 x(0;a) )two adja ent os illators from in-phase syn hronization to
1 3
anti-phase syn hronization or from anti-phase syn hroniza+"(y(0;a)y):3 (0;a)tion to in-phase syn hronization. The phase-inversion wave
behaviors observed propagation, penetration, ree tion, and [Cornerbottom℄ (left:(a; b)=(0; 1). right:(a; b)=(N; N 1).)
disappearan e. This paper laries disappearan e me hanismdx M a(3)
between two phase-inversion waves, and ree tion me had = y( M;a) ;nism when two phase-inversion waves arrive at a orner on ady M alatti e of oupled os illators system at same time.
d = x( M;a) + ( x( M 1;a) + x( M;b)
1. Introdu tion
2 x( M;a) ) + "(y( M;a) 31 y3(M;a) ):A lot of syn hronization phenomena an be observed andstudied in various elds[1℄-[2℄. For example, the laser is high[Center℄ (0 < k < M . 0 < l < N .)intensity ray bundle, the syn hronization of reies, and sodx k l(4)
on. In our previous study, we observed and analyzed synd = y(k;l) ;hronization phenomena between adja ent os illators on oudy k lpled os illators system as a ladder and a latti e[3℄-[5℄. Wed = x(k;l) + ( x(k+1;l) + x(k 1;l) + x(k;l+1) + x(k;l 1)
dis overed the phase-inversion wave, whi h phenomena of
4 x(k;l) ) + "(y(k;l) 13 y3(k;l) ):hanging phase states between two adja ent os illators fromin-phase syn hronization to anti-phase syn hronization orfrom anti-phase syn hronization to in-phase syn hronization. [Edge℄(top:(a; b)=(0; 1).bottom:(a; b)=( M; M 1).both:0< l <N .)We analyzed propagation me hanism of the phase-inversion
wave, and ree tion me hanism between two phase-inversion
dx a lwaves.
(5)d = y(a;l) ;In this paper, we larify disappearan e me hanism between
dy a ltwo phase-inversion waves, and ree tion me hanism whend = x(a;l) + ( x(a;l 1) + x(a;l+1) + x(b;l) 3 x(a;l) )two phase-inversion waves arrive at a orner at same time+ "(y(a;l) 13 y3(a;l) ):using instantaneous frequen y of ea h os illators and phase
di eren es between adja ent os illators on the latti e system.
(left:(a; b)=(0; 1). right:(a; b)=(N; N 1). both:0<k< M .)
(0; )

(0; )

(

;

)

(

;

)

( ; )

( ; )

(

; )

(

; )

dx k a2. Cir uit model(6)d = y(k;a) ;A lot of van der Pol os illators are oupled by indu tors L0
dy k aas a latti e(see Fig. 1). The number of olumn and the numd = x(k;a) + ( x(k 1;a) + x(k+1;a) + x(k;b) 3 x(k;a) )ber of row of this system are assumed as  M + 1 and N + 1
respe tively. We name ea h os illator OSC(k,l). A voltage+ "(y(k;a) 13 y3(k;a) ):of ea h os illator is named v(k;l) , and a urrent of indu tor ofea h os illator is named i(k;l) (see Fig. 1). The ir uit equa- Theorresponds to the oupling parameter. The " orretions of this ir uit model are normalized by Eq. (1), and the sponds to the nonlinearity of ea h os illator. This system
normalized ir uit equations are shown as Eqs. (2)(6).
is simulated by the fourth order Runge-Kutta method using
q Cg
qg
Eqs. (2)-(6). The phase-inversion waves are shown in Fig. 2.
x(k;l) ; v(k;l) =y(k;l) ;i(k;l) =3Lg
3g
qL(1) Figure 2A expresses attra tor of ea h os illators. Figure 2
pdL
B expresses itineran y of phase di eren e sum of voltages of
LC ;
t =
d =   ; = L ; " = g1 C :adja ent os illators is shown along the time.
( ; )

( ; )

1

1

3

3

0

- 105 -

L0

(0,N)

L0

OSC

OSC

L0
(1,1)
(1,0)

OSC

OSC

OSC

i(k,l)

L0

OSC

L C

v(k,l)[degree/cycle]

(0,1)

L0

OSC

anti-phase

(0,0)

4
3
2
1
0

(M,0)

OSC

OSC

OSC

in-phase

L0(M,N)

Figure 1: Cir uit Model.

1
2
3

0

-20

-40

-60

-80

-100

-120

-140

-160

-180

[degree]

Initial phase differences
A B

Figure 3: Attra ting for es.
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

Figure 2: The phase-inversion waves on 20x20 os illators(A:an attra tor of ea h os illator( urrent vs. voltage),
B:a sum of voltages of adja ent os illators(sum of voltage Figure 4: The dete tion method of frequen ies and the phase
di eren es.
vs. time)).
Time

3. Attra ting for e
Attra ting for es to in-phase or anti-phase syn hronization are investigated on one parameter-set where the phaseinversion waves an be observed(see Fig. 3). An equation of
the instantaneous frequen y of OSC(k, l) is obtained as follows. The instantaneous frequen y is named f(k;l) (a) wherea expresses the number of times of the peak value of thevoltage. Time of a peak value of the voltage of OSC(k, l)
is assumed as (k;l) (a)(see Fig. 4). Similarly, (k+1;l) (a) and(k;l+1) (a) are de ided. The f(k;l) (a) is obtained by Eq. (7).
( )=f(k;l) a1

 k;l (a)  k;l (a 1)
(

)

(

)

:

(7)

The phase di eren e is al ulated as follows. A phase differen e between OSC(k, l) and OSC(k + 1, l) and a phase
di eren e between OSC(k, l) and OSC(k, l + 1) are obtained. The phase di eren es are assumed as (k;l)(k+1;l) (a)and (k;l)(k;l+1) (a) respe tively(see Fig. 4). The (k;l)(k+1;l) (a)and (k;l)(k;l+1) (a) are obtained by Eq. (8).
)
 k l k+ l (a) =  k l(a(a) )  k+(al (a1)
 360; [degree℄( ; )( ; )(

1; )

(k;l)

(

1; )

(k;l)

)
 k l k l+ (a) =  k l(a(a) )  k l(+a (a1)
 360: [degree℄:( ; )( ; )( ;

1)

(k;l)

( ;

1)

(8)

(k;l)

Attra ting for es are observed as follows:
1. The phase di eren es between all adja ent os illators
are xed to arbitrary values as initial value in this latti e system.
2. The phase di eren e between OSC(1,4) and OSC(1,5)
after one y le from the initial value is analyzed hanging the initial phase di eren e.
A verti al axis of Fig. 3 expresses a variation of phase differen e per one y le. An upper dire tion shows attra tingfor e to anti-phase. A lower dire tion shows attra ting for e
to in-phase. A horizontal axis shows initial phase di eren es.
Therefore, the length of line shows a strength of attra tingfor es at ea h initial phase di eren e. Attra ting for e to inphase is the strongest when the initial phase di eren e is -40
degrees. Attra ting for e to anti-phase is the strongest when
the initial phase di eren e is -130 degrees.
4. Behavior of phase-inversion waves
We an observe some behaviors of phase-inversion waves
on above systems. These behaviors are a propagation, aree tion at an edge, a ree tion between phase-inversion
waves, a penetration, a ree tion at a orner, distin tion anda disappearan e. Moreover, these behaviors an be lassied by frequen ies, be ause three frequen ies are observedin steady states. The syn hronizations for verti al dire tionand for horizontal dire tion needs to be onsidered, be ause
this system is 2 dimensional array. Therefore, three type synhronizations 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(k1, l): in-phase
syn hronization.
2. f(OSC(k, l)OSC(k, l + 1) and OSC(k, l)OSC(k, l 1):
in-phase syn hronization. OSC(k, l)OSC(k + 1, l), andOSC(k, l)OSC(k 1, l): anti-phase syn hronization.g or
fOSC(k, l)OSC(k, l + 1), and OSC(k, l)OSC(k, l 1):
anti-phase syn hronization. OSC(k, l)OSC(k + 1, l),and OSC(k, l)OSC(k + 1, l): in-phase syn hronization.g
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
syn hronization.
In this paper, we all the 1st type syn hronization in-andin-phase syn hronization. The 2nd type syn hronizationis alled in-and-anti-phase syn hronization. The 3rd typesyn hronization is alled anti-and-anti-phase syn hronization. An ea h instantaneous frequen y f(k;l) of OSC(k, l) is
obtained in ea h syn hronization type. In the 1st situational
syn hronization, f(k;l) is fin in . In the 2nd situational syn hronization, f(k;l) is fin anti . In the 3rd situational syn hroniza-- 106 -

Table 1: Chara teristi s of the phase-inversion waves.
Names of behaviors

Itineran ies of instan-

fin inPropagations

,
,
&
,
,
,
,
&
,
, middle offin anti

fin anti

Penetrations

fin infin inRee tions at an edge
fin infanti anti
fin antiRee tions between twophase-inversion waves
Disappearan e

fin anti

fin infin inTwo phase-inversion waves arrives at an os illator from verti al dire tion and horizontal dire tion, and ea h phase-inversion wave penetrates ea h other.
When a phase-inversion wave arrives at an edge, the phase-inversion wave ree ts and propagates to where they ame from. Sometime this phenomenon is happened with penetration.

fanti anti

and fanti anti

,
&
,
,

fin anti

The phase-inversion waves propagate to verti al dire tion or horizontal dire tion. The verti alphase-inversion waves move from the horizontal phase-inversion waves independently.

fanti anti

fin antiRee tions at a orner

Phenomena

taneous frequen ies

fin anti,

fanti anti

fanti anti

When two phase-inversion waves oming from the verti al dire tion and the horizontal dire tion arrive at a orner os illator at the same time, the phase-inversion waves ree t andpropagate to where they ame from.
When two phase-inversion waves oming from the opposite dire tions arrive to two adja ent
os illator at same time, the phase-inversion waves ree t and propagate to where they ame
from.
When two phase-inversion waves oming from the opposite dire tions arrive to one os illatorat same time, the phase-inversion waves disappear.

Instantaneous Frequency

f anti-anti5f(1,0)
andf(0,1)

f in-anti

45f(2,0)
andf(0,0)

f(0,2)
2f in-in

7

Time[ ](a) Instantaneous frequency

tion, f(k;l) is fanti anti . These behaviors are shown in Table 1.

360

Phase Difference [degree]

Figure 5: Disappearan e of the phase-inversion waves on
11x11 os illators.

(0,0)(1,0)and(0,0)(0,1)6

180(2,0)(3,0)and(0,2)(0,3)(1,0)(2,0)and(0,1)(0,2)

5. Me hanism
0
We analyze the ree tion at a orner and the disappearan e.
Time[ ](b) Phase difference
The phase-inversion wave shows in Fig. 2 and Fig. 5. The
me hanisms are made lear using instantaneous frequen y of Figure 6: Transitions of phase di eren e and frequen ies by
ea h os illator and phase di eren e between adja ent os il- ree tion at a orner of two phase-inversion waves.
lators. The oupling parameter is xed as = 0:01, and nonlinearity is xed as " = 0:05.
5.1. Ree tion me hanism at a orner
We observe the ree tion between two phase-inversion
waves whi h are arrives at a orner os illator at the same timefrom the verti al dire tion and the horizontal dire tion. A
me hanism of ree tion at a orner is shown in Tab. 2 andTime[ ](a) Instantaneous frequency
Fig. 6. In Fig. 6(a), the verti al axis is instantaneous frequen360
ies, and horizontal axis is time. In Fig. 6(b),the verti al axis
is the phase di eren es, and the horizontal axis is the time.
180
5.2. Disappearan e me hanism
0
We an observe the disappearan e. A me hanism of disappearan e is shown in Tab. 3 and Fig. 7. In Fig. 7(a), the-180
verti al axis is instantaneous frequen ies, and horizontal axis-360
is time. In Fig. 7(b), the verti al axis is the phase di eren es,
Time[ ]and the horizontal axis is the time.
(b) Phase difference
Figure 7: Transitions of phase di eren e and frequen ies by
6. Con lusion
We observed and analyzed ree tion me hanism when disappearan e between two phase-inversion wave.
two phase-inversion waves arrive at a orner on a latti e1

3

Instantaneous Frequency

f anti-anti

f in-anti13f(5,0)

f(4,0)

andf(6,0)

f in-in7

5

Phase Difference [degree]

62(4,0)(5,0)(6,0)(7,0)

42(3,0)(4,0)(5,0)(6,0)

6- 107 -

Table 2: Ree tion me hanism of two phase-inversion waves at a orner(see Fig. 6).
no.

verti al dire tion

horizontal dire tion0

Now the phase state between adja ent os illators of the verti al dire - Now the phase state between adja ent os illators of the horizontal dire tion is the in-phase syn hronization. The phase-inversion waves, whi h tion is the in-phase syn hronization. The phase-inversion waves, whi hhange from in-phase syn hronization to anti-phase syn hronization,
hange from in-phase syn hronization to anti-phase syn hronization,
are arrived at OSC(2,0) from OSC(5,0).
are arrived at OSC(0,2) from OSC(0,5).
1
(1 0)(2 0) starts to in rease toward 180 degrees from 0 degree, be ause (0 1)(0 2) starts to in rease toward 180 degrees from 0 degree, be ausef(2 0) starts to in rease from fin in toward fin anti .
f(0 2) starts to in rease from fin in toward fin anti .
2f(1 0) starts to in rease from fin in toward fin anti , be ause (1 0)(2 0)f(0 1) starts to in rease from fin in toward fin anti , be ause (0 1)(0 2)starts to in rease toward 180 degrees from 0 degree.
starts to in rease toward 180 degrees from 0 degree.
3
(0 0)(1 0) starts to in rease toward 180 degrees from 0 degree, be ause (0 0)(0 1) starts to in rease toward 180 degrees from 0 degree, be ausef(1 0) starts to in rease from fin in toward fin anti .
f(0 1) starts to in rease from fin in toward fin anti .
4f(1 0) arrives around fin anti , be ause (1 0)(2 0) arrives around 180 def(0 1) arrives around fin anti , be ause (0 1)(0 2) arrives around 180 degrees. f(1 0) starts to in rease from fin anti toward fanti anti .
grees. f(0 1) starts to in rease from fin anti toward fanti anti .
5f(0 0) arrives around fin anti , be ause (0 0)(1 0) and (0 0)(0 1) arrive around 180 degrees at the same time. (0 0)(1 0) and (0 0)(0 1) start to in rease
toward 360 degrees from 180 degrees. The orner os illator an not stable at anti-phase syn hronization. Therefore, f(0 0) arrives around middleof fin in and fanti anti . f(1 0) , f(2 0) , f(0 1) , and f(0 2) arrive around middle of fin anti and fanti anti . Therefore, f(0 0) starts to de rease toward fin in
again. f(1 0) , f(2 0) , f(0 1) , and f(0 2) start to de rease toward fin anti again.
6
(1 0)(2 0) starts to in rease toward 360 degrees from 180 degrees.
(0 1)(0 2) starts to in rease toward 360 degrees from 180 degrees.
7
f(0 0) arrives around fin in .
Two phase-inversion waves ree t at a orner by this me hanism.
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Table 3: Disappearan e me hanism(see Fig. 7).
no.

upper side

lower side0

Now the phase state between adja ent os illators of the horizontal dire tion is the in-phase syn hronization. In verti al dire tion, the phaseinversion waves, whi h hange from anti-phase syn hronization to in-phase syn hronization, are arrived at OSC(3,0) and OSC(7,0).
1
f(4 0) starts to de rease from fin anti toward fin in , be ause (3 0)(4 0)f(6 0) starts to de rease from fin anti toward fin in , be ause (6 0)(7 0)starts to in rease toward 0 degree from -180 degrees.
starts to de rease toward 0 degree from 180 degrees.
2(4 0)(5 0) starts to in rease toward 360 degrees from 180 degrees, be- (5 0)(6 0) starts to de rease toward -360 degrees from -180 degrees, beause f(4 0) starts to de rease from fin anti toward fin in .
ause f(6 0) starts to de rease from fin anti toward fin in .
3
f(5 0) starts to de rease from fin anti toward fin in , be ause (4 0)(5 0) and (5 0)(6 0) start to hange toward plus/minus 360 degrees fromplus/minus 180 degrees.
4(3 0)(4 0) arrives around 0 degree, be ause f(3 0) arrives around fin in . (6 0)(7 0) arrives around 0 degree, be ause f(7 0) arrives around fin in .
The attra ting for e an not exist around 180 degrees. Therefore, 180 The attra ting for e an not exist around 180 degrees. Therefore, 180
degrees is easy to stable.
degrees is easy to stable.
5f(4 0) arrives around fin in , be ause (3 0)(4 0) arrives around 0 degree.
f(6 0) arrives around fin in , be ause (6 0)(7 0) arrives around 0 degree.
6
(4 0)(5 0) arrives around 360 degrees, be ause f(4 0) arrives around (5 0)(6 0) arrives around -360 degrees, be ause f(6 0) arrives around
fin in . The attra ting for e an not exist around 0 degree. Therefore,
fin in . The attra ting for e an not exist around 0 degree. Therefore,
0 degree is easy to stable.
0 degree is easy to stable.
7f(5 0) arrives around fin in , be ause (4 0)(5 0) and (5 0)(6 0) arrives around plus/minus 360 degrees.
8
Ea h frequen y xes fin in , the phase di eren es x in-phase syn hronization.
Two phase-inversion waves disappear by above me hanism.
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of oupled os illators systems. Furthermore, we observed
Periodi Phases in Populations of Nonlinearly Couand analyzed disappearan e me hanism between two phasepled Os illators with Bimodal Frequen y Distributions,
inversion waves. These me hanisms are made lear using inPhysi a D: Nonlinear Phenomena vol. 113, issues 1, pp.
stantaneous frequen y of ea h os illator and phase di eren e
7997, Feb. 1998.
between adja ent os illators.
[3℄ Y. Setou, Y. Inami, Y. Nishio and A. Ushida On a Ladderof Os illators Sharing Indu tors, Pro . of NOLTA'97,
A knowledgements
vol. 1, pp. 565-568, Nov. 1997.
This resear h is supported by the Grants-in-Aid for Young
S ienti Resear h (B) (No. 19760270) from the Japan So i- [4℄ M. Yamau hi, M. Okuda, Y. Nishio and A. Ushida,
ety for the Promotion of S ien e.
Analysis of Phase-Inversion Waves in Coupled Osillators Syn hromizing at In-and-Anti-Phase, IEICE
Referen es
Trans. Fundamentals, vol.E86-A, no.7, pp.1799-1806,[1℄ R. E. Mirollo and S. H. Strogatz Syn hronization of
July. 2003.
Pulse-Coupled Biologi al Os illators, SIAM Journal on[5℄ H. Aburatani, S. Yamane, M. Yamau hi and Y. Nishio
Applied Mathemati s, vol. 50, no. 6, pp. 1645-1662, De .
Ree tion Me hanisms of a Phase-Inversion Wave at
1990.
an Edge of 2D Latti e Os illators, Pro . NOLTA 2009,[2℄ L. L. Bonilla, C. J. Pérez Vi ente and R. Spigler, Timepp.175-178, O t. 2009.

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