2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Phase-Inversion Waves in Simultaneously Existing
Two Synchronization Modes of 2D Oscillator Network
Yoshihito Todani† , Masayuki Yamauchi† and Yoshifumi Nishio‡†Department of Electronics and Computer Engineering,
Hiroshima Institute of Technology
2-1-1,Miyake,Saeki-ku,Hiroshima,Japan
Email: m111307@cc.it-hiroshima.ac.jp‡Department of Electrical and Electronic Engineering,
Tokushima University
2-1,Minami-Josanjima,Tokushima,Japan
Abstract—We have been observing synchronization phenomena on coupled oscillators systems. We can observe
special phenomena on lattice oscillators by using computer
simulations. The special phenomena is the synchronization
states in in-and-anti-phase synchronizations for vertical direction and in-phase synchronizations for horizontal direction. The synchronization states called two synchronization
modes. In this paper, we investigate and analyze the phaseinversion waves on two synchronization modes on 2D oscillator networks. We separate the observation phenomena for 5
regions, and propagation mechanisms of the phase-inversion
waves are analyzed.
1. Introduction
Synchronization phenomena can be observed in the biological bodies, the atomic world, the outer space, the mechanical systems, the electrical circuits, and so on. Especially, it is easy that synchronization phenomena are observed on coupled oscillators systems which are electricalcircuits. Therefore, many phenomena have been reported
by using electrical circuits[1]. When van der Pol oscillators
are coupled by inductor as a ladder or as a 2D lattice, theall oscillators are an in-phase synchronization or an in-andanti-phase synchronization. It is called the in-and-anti-phasesynchronization that the in-phase synchronizations and theanti-phase synchronizations alternately exist. The in-andanti-phase synchronization can be stable when phase states
between edge oscillators and next oscillators are anti-phasesynchronizations. On 2D oscillator network, we can observe that vertical synchronization states diﬀer from horizontal synchronization states[2]. For example, if the verticalsynchronization states are the in-phase synchronizations, thehorizontal synchronization states are the in-and-anti-phasesynchronizations. The synchronization states are called two
synchronization modes.
In our previous study, we discovered a special wave motion that a phase state continuously propagates on a ladderoscillator networks or on 2D oscillator networks. Especially,
we discovered a wave motion which switches phase states
between adjacent oscillators and continuously exists. The
wave motion is called phase-inversion waves[3].
In this study, we investigate and analyze the phaseinversion waves on two synchronization modes on 2D oscil-

(0,0)

L0

OSC

L0

L0

OSC

L0
(1,0)

(0,1)

OSC

(1,1)

(0,N-1)
OSC

L0

L0
L0

L0

OSC

OSC

i(k,l)(N-1,0)

L0

L0

L0

L0

L0

L0
OSC

OSC

OSC

L C

L0

L0
OSC

(N-1,N-1)

Figure 1: Circuit model.
lator networks. We separate the observation phenomena for5 regions. Furthermore, propagation mechanisms of the horizontal and the vertical phase-inversion waves are analyzed.
2. Circuit model
The van der Pol oscillators are coupled by inductors L0 as
a 2D lattice (see Fig. 1) . The numbers of column and rowof this system are assumed as “N” respectively. We nameeach oscillator OSC(k,l)(0 ≤ k and l ≤ N − 1). A voltageof each oscillator is named v(k,l) , and a current of a inductorin each oscillator is named i(k,l) (see Fig. 1) . An equation of
the nonlinear negative resistor is shown as Eq. (1). Circuit
equations are normalized by Eq. (2). The normalized circuit
equations are shown as Eqs. (3)–(7). The α corresponds to acoupling parameter. The ε corresponds to a nonlinearity ofeach oscillator. This circuit is simulated by using the fourthorder Runge-Kutta method and Eqs. (3)–(7).
ir (v(k,l) ) = −g1 v(k,l) + g3 v3(k,l) .


Cg1
g1
x
,
v
=
i(k,l) = 3Lg
(k,l)
(k,l)
3g3 y(k,l) ,3√dt = LCτ, dτ= “ · ”, α = LL0 , ε = g1 CL .

(1)
(2)

[Corner–top] (left:(a, b)=(0, 1), right:(a, b)=(N − 1, N − 2).)dx(0,a)(3)dτ = y(0,a) ,dy(0,a)dτ = −x(0,a) + α(x(0,b) + x(1,a) − 2x(0,a) )+ ε(y(0,a) − 13 y3(0,a) ).
[Corner–bottom] (left:(a, b)=(0, 1), right:(a, b)=(N − 1, N − 2).)dx(N,a)(4)dτ = y(N,a) ,- 361 -v(k,l)X

Y

dy(N,a)dτ = −x(N,a) + α(x(N−1,a) + x(N,b) − 2x(N,a) )+ ε(y(N,a) − 13 y3(N,a) ).
[Center] (0 < k < N − 1, 0 < l < N − 1.)dx(k,l)dτ = y(k,l) ,(5)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−1, N−2),both:0 < l < N−1.)dx(a,l)dτ = y(a,l) ,Figure 2: Computer simulation result of the phase-inversion
waves in the in-and-anti-phase synchronization for verticaldirection and the in-phase synchronization for horizontal direction
on 9×9 2D oscillator.
Y
X

(6)

dy(a,l)dτ = −x(a,l) + α(x(a,l−1) + x(a,l+1) + x(b,l) − 3x(a,l) )+ ε(y(a,l) − 13 y3(a,l) ).
(left:(a, b)=(0, 1), right:(a, b)=(N −1, N −2), both:0 < k < N −1.)dx(k,a)dτ = y(k,a) ,(7)dy(k,a)dτ = −x(k,a) + α(x(k−1,a) + x(k+1,a) + x(k,b) − 3x(k,a) )+ ε(y(k,a) − 13 y3(k,a) ).
3. Phase-invertion waves
Phase-inversion waves in the in-and-anti-phase synchronization for vertical direction and the in-phase synchronization for horizontal direction are shown in Figs. 2 and 3. The
Figure 2 is observed when N is nine(an odd number). Signs
of initial values are shown in Fig. 4. “X” expresses an attractor of each oscillator(current vs. voltage). “Y” expresses anitinerancy of phase diﬀerence by which sum of voltages ofadjacent oscillators is shown along the time(sum of voltages
vs. time). Black areas are almost the in-phase synchronization. White areas are almost the anti-phase synchronization.
In each figure, we can observe phase-inversion waves which
simultaneously propagate to horizontal direction in the inphase synchronization and to vertical direction in the in-andanti-phase synchronization. The Figure 3 is observed when
N is ten(an even number). Signs of initial values are shownin Fig. 5. In this study, basic synchronization states of vertical direction are set as the in-and-anti-phase synchronizationand synchronization states of horizontal direction are set as
the in-phase synchronization.
3.1. Regions
We separate the observation phenomena for five regions.
The coupling parameter α is changed from 0.010 to 0.30, every 0.010. The nonlinearity ε is changed from 0.010 to 0.50,
every 0.010. A wave which doesn’t disappear before 30000τ
is assumed as the phase-inversion wave in this paper. The
Figure 6 shows regions which the phase-inversion waves canbe observed when N equals 9. The Figure 7 shows regions
which the phase-inversion wave can be observed when N
equals 10. The vertical and horizontal phase-inversion waves

Figure 3: Computer simulation result of the phase-inversion
waves in the in-and-anti-phase synchronization for verticaldirection and the in-phase synchronization for horizontal direction on 10×10 2D oscillator.
can be observed in region(i)(see Figs. 6 and 7). The vertical phase-inversion waves can be observed and the horizontal phase-inversion waves can’t be observed in region(ii)(seeFig. 6). The vertical phase-inversion waves and the complex phenomena for horizontal direction can be observed inregion(iii)(see Figs. 6 and 7). The vertical phase-inversion
waves and the synchronization phenomena without waves forhorizontal direction can be observed in region(iv)(see Fig. 7).
The complex phenomena for vertical direction and horizontal direction can be observed in region(v)(see Figs. 6 and 7).
3.2. Mechanisms
We can observe a phenomenon that vertical phaseinversion waves and horizontal phase-inversion waves propagate at the same time. We analyze the propagation mechanism of phase-inversion waves by using an instantaneous
frequency of each oscillator and phase diﬀerences between
adjacent oscillators. The coupling parameter is fixed as
α=0.010, and nonlinearity is fixed as ε=0.150. The instantaneous frequency is defined as Eq. (8).
1,(8)
τ(k,l) (a) − τ(k,l) (a − 1)
where “a” expresses the number of times of the voltage positive peak value. τ(k,l) (a) means the time of OSC(k,l)(seeFig. 8). Similarly, τ(k+1,l) (a) and τ(k,l+1) (a) are defined. Somefrequencies are observed in this phenomena. These frequencies are needed to consider the synchronization states forthe vertical direction and the horizontal direction, because
this system is 2-dimensional array. Five phase states have

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f(k,l) (a) =









0.5
0.45

Nonlinearity ε

0.4

Figure 4: Sign of initial value of each oscillator of Fig. 2.

(v)

0.35
0.3
0.25
0.2
0.15












(iii)

(i)

0.1
0.05
0

(ii)
0

0.05

0.1

0.15

0.2

0.25

0.3

Coupling parameter α

Figure 5: Sign of initial value of each oscillator of Fig. 3.

Figure 6: Regions of when N=9.
0.5

to be considered for an oscillator of 2D lattice, because thein-phase synchronization and the anti-phase synchronization
are exist in the same time. Therefore, five types synchronization states are observed in 2D lattice as follows:

0.45
0.4

Nonlinearity ε

1. fiiii: Four phase states are the in-phase synchronization.

(i)
(iii)
0

0.05

0.1

0.15

0.2

0.25

0.3

Coupling parameter α

Figure 7: Regions of when N=10.
(k, l)(a)V

OSC(k, l)

a-1

a

a+1

Time

(k+1, l)(a)V

5. faaaa: Four phase states are the anti-phase synchronization.

OSC(k+1, l)

a-1

a

a+1

Time

Figure 8: The calculation method of the instantaneous frequencies and the phase diﬀerences.
Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 25730152.

Φ(k,l)(k+1,l) (a) =References(9)

4. Conclusion
We analyzed the phase-inversion waves on simultaneously
existing two synchronization modes on 2D oscillator networks. We clarified regions of phenomena with the phaseinversion waves in the in-and-anti-phase synchronization forvertical direction and the in-phase synchronization for horizontal direction. Furthermore, the propagation mechanismsof the phase-inversion waves were analyzed.

0.2
0.15

0

4. fiaaa: Three of four phase states are the anti-phase synchronization. Another phase state is the in-phase synchronization.

Propagation mechanism of the phase-inversion wave isshown in Table 1(see Fig. 9), and the propagation mechanism of the horizontal phase-inversion wave is shown in Table 2(see Fig. 10)

0.3
0.25

0.05

3. fiiaa: Two of four states are the in-phase synchronization. Other phase states are the anti-phase synchronization.

τ(k,l) (a) − τ(k+1,l) (a)× 2π [rad]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) =× 2π [rad].
τ(k,l) (a) − τ(k,l) (a − 1)

(iv)

0.1

2. fiiia: Three of four phase states are the in-phase synchronization. Another phase state is the anti-phase synchronization.

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 diﬀerence are assumed as Φ(k,l)(k+1,l) (a)The Φ(k,l)(k+1,l) (a) andand Φ(k,l)(k,l+1) (a), respectively.
Φ(k,l)(k,l+1) (a) are obtained by Eq. (9) (see Fig. 8).

(v)

0.35

[1] T. Endo and S. Mori, “Mode Analysis of TwoDimensional Low-Pass Multimode Oscillator,” IEEE
Trans. Circuits and Syst., vol. 23, no. 9, pp. 517-530,
Sept. 1976.
[2] S. Yamane, M. Yamauchi and Y. Nishio, “Classificationof phenomena on coupled oscillators system as a lattice,”
Proc. NCSP’09, pp.249-252, Mar. 2009.
[3] M. Yamauchi, M. Okuda, Y. Nishio and A. Ushida,“Analysis of Phase-Inversion Waves in Coupled Oscillators Synchronizing at In-and-Anti-Phase,” IEICE Trans.
Fundamentals, vol.E86-A, no.7, pp.1799-1806, July.
2003.

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Table 1: Propagation mechanism of a phase-inversion wave
in a column (see Fig. 9).

Table 2: Propagation mechanism of a phase-inversion wave
in a row (see Fig. 10).

no.

Mechanism0

At first, Φ(3.3)(4,3) is −π, Φ(4,3)(5,3) , Φ(4,2)(4,3) , Φ(4,3)(4,4) ,no.

MechanismΦ(5,2)(5,3) and Φ(5,3)(5,4) are 0, and Φ(5.3)(6,3) is π. The vertical phase-inversion wave comes from 0th row to 3rd row in the0Φ(2,13)(2,14) , Φ(2,14)(2,15) , Φ(2,15)(2,16) , Φ(1,14)(2,14) andΦ(1,15)(2,15) are 0. Φ(2,14)(3,14) and Φ(2,15)(3,15) are π. The
horizontal phase-inversion wave comes from 11th column toin-and-anti-phase synchronization in each column.
12

Φ(3,3)(4,3) starts to change from −π toward −2π by the vertical

13th column in the in-phase synchronization in each row.

phase-inversion wave.

1

f(4,3) starts to change from fiiia to fiiii , because Φ(3.3)(4,3)
is changing from −π to −2π and Φ(4,2)(4,3) , Φ(4,3)(4,4) and2

Φ(4,3)(5,3) are 0.
3

Φ(4,3)(5,3) starts to change from 0 to π, because f(4,3) is changing from fiiia to fiiii , and f(5,3) is not changed from fiiia yet.

Φ(2,14)(3,14) don’t change, because f(1,14) , f(2,14) , and f(3,14) aresimilarly changed by each horizontal phase-inversion wave.

are changing from fiiia to fiiii at same time by each verticalphase-inversion wave.

3

f(5,3) starts to change from fiiia toward fiiaa , because Φ(4,3)(5,3)

4is changing from 0 to π, Φ(5,2)(5,3) and Φ(5,3)(5,4) are 0, andΦ(5,3)(6,3) is π.
5Φ(5,3)(6,3) starts to change from π toward 0, because f(5,3) is5

f(2,14) continuously changes toward fiaaa , because Φ(2,13)(2,14)f(2,15) starts to change from fiiia toward fiiaa , because
Φ(2,14)(2,15) is changing from 0 to −π, Φ(2,15)(2,16) andΦ(1,15)(2,15) are 0 and Φ(2,15)(3,15) is π.

f(4,3) starts to change toward fiiia again without arriving at fiiii ,because Φ(3,3)(4,3) is changing to 2π, Φ(4,3)(5,3) is changing toπ, and Φ(4,2)(4,3) and Φ(4,3)(4,4) is 0. So, three of four phase

6

states around OSC(4,3) are changing toward the in-phase synchronization state, and another phase state is changing toward7

Φ(2,15)(2,16) starts to change from 0 toward π, because f(2,15) ischanging from fiiia to fiiaa .
f(2,15) continuously changes toward fiaaa , because Φ(2,14)(2,15)
and Φ(2,15)(2,16) are changing from 0 to π, Φ(1,15)(2,15) is 0 andΦ(2,15)(3,15) is π.

anti-phase synchronization.
7

Φ(2,14)(2,15) starts change from 0 toward −π, because f(2,14) ischanging from fiiia to fiiaa .
and Φ(2,14)(2,15) are changing from 0 to −π, Φ(1,14)(2,14) is 0 andΦ(2,14)(3,14) is π.

changing from fiiia to fiiaa .
6f(2,14) starts to change from fiiia toward fiiaa , because
Φ(2,13)(2,14) is changing from 0 to −π, Φ(1,14)(2,14) andΦ(2,14)(2,15) are 0 and Φ(2,14)(3,14) is π. Φ(1,14)(2,14) andΦ(4,2)(4,3) and Φ(4,3)(4,4) are 0, because f(4,2) , f(4,3) , and f(4,4)

4Φ(2,13)(2,14) starts to change from 0 toward −π by the horizontal
phase-inversion wave.

Φ(2,13)(2,14) arrives at −π.

f(5,3) starts to change toward fiiia again without arriving at fiiaa ,8because Φ(4,3)(5,3) is changing to π, Φ(5,3)(6,3) is changing to 0,
and Φ(5,2)(5,3) and Φ(5,3)(5,4) is 0.

9f(2,14) arrives at fiaaa .

10Φ(2,14)(2,15) arrives at −π.

8Φ(3,3)(4,3) arrives at −2π.

11f(2,15) arrives at fiaaa .

9f(4,3) arrives at fiiia again.

12Φ(2,15)(2,16) arrives at −π.

10Φ(4,3)(5,3) arrives at π.

11f(5,3) arrives at fiiia again.

12Φ(5,3)(6,3) arrives at 0.

The horizontal phase-inversion wave
propagate by this mechanism.

The vertical phase-inversion wave
propagate by this mechanism.
f iaaa7f(5,3)

11

4

f iiia2

Instantaneous Frequency

Instantaneous Frequency

f iiaa9f(4,3)

6

9

f iiaaf(2,14)f(2,14)

4
2

f iiiaPhase Difference [rad]Phase Difference [rad]105(5,3)(6,3)(4,3)(5,3)3

12(4,3)(4,4)(4,2)(4,3) andand(5,2)(5,3) and(5,3)(5,4)

-π
1

-2π

5

π

Time[ ]
Instantaneous frequency

07

Time[ ]
Instantaneous frequency

f iiiiπ11(3,3)(4,3)(2,15)(3,15) and(2,14)(2,15)

0

6(1,14)(2,14) and(1,15)(2,15)

1(2,15)(2,16)(2,13)(2,14)

-π

8

3

(2,14)(3,14)

8

10

12

Time[ ]
Phase difference

Time[ ]
Phase difference

Figure 9: Transitions of frequencies and phase diﬀerences by
propagation of a phase-inversion wave in columns.

Figure 10: Transitions of frequencies and phase diﬀerencesby propagation of a phase-inversion wave in rows.

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