2013 International Symposium on Nonlinear Theory and its Applications
NOLTA2013, Santa Fe, USA, September 8-11, 2013

Analysis of Phase-Inversion Waves in In-and-Anti-Phase Synchronization
on Coupled Van der Pol Oscillators as a 2D Lattice by Using an Actual Circuit
Yoshihito Todani† , Seiko Kunihiro†, Masayuki Yamauchi† and Yoshifumi Nishio‡†Department of Electronics and Computer Engineering,
Hiroshima Institute of Technology
2-1-1,Miyake,Saeki-ku,Hiroshima,Japan
Email: todani4411@gmail.com‡Department of Electrical and Electronic Engineering,
Tokushima University
2-1,Minami-Josanjima,Tokushima,Japan
Abstract—
In this paper, we observe phase-inversion waves which simultaneously propagate to horizontal direction and verticaldirection in in-and-anti-phase synchronization by using actual circuit experiments and numerical experiments. We observe transitions of propagation velocities by changing coupling parameters between two adjacent oscillators. We clarify a reflection mechanism of the phase-inversion waves at acorner when there are in-and-anti phase synchronization forvertical direction and in-phase synchronization for horizontal
direction.
1. Introduction
Synchronization phenomena are studied by many researchers. The synchronization phenomena are observed ateverywhere of the natural world, and the phenomena are used
for human society. For example, the phenomena are used tocommunicating system, laser, and so on[1]-[2]. The synchronization phenomena can be observed in systems including
many oscillators very often and it is diﬃcult to send the social lives without the phenomena.
In our previous study, we have been observing synchronization phenomena on a ladder networks and 2D lattice networks of coupled van der Pol oscillators[3]. We observed
not only in-phase, anti-phase or in-and-anti-phase synchronization of steady states but also phenomena of propagating phase diﬀerences. In-and-anti-phase synchronization isa phase state which in-phase synchronization and anti-phasesynchronization alternately exist. Especially, we discovereda wave which switches phase states between adjacent oscillators and continuously exists. The wave is called phaseinversion wave. We observed the phase-inversion waves
which simultaneously propagate to horizontal direction andvertical direction in in-and-anti-phase synchronization on the
2D lattice shape networks by using a computer simulation.
However, the phase-inversion waves which propagates to the
double directions were not able to be observed by using actual circuit experiments.
In this paper, we observe phase-inversion waves which simultaneously propagate to horizontal direction and verticaldirection in the in-and-anti-phase synchronization by using
actual circuit experiments and numerical experiments. Propagation velocities are investigated when coupling parameters

between two adjacent oscillators are changed. A reflection
mechanism of the phase-inversion waves at a corner is clarified in in-and-anti-phase synchronization for vertical direction and in-phase synchronization for horizontal direction.
2. Circuit model
The van der Pol oscillators are coupled by inductors L0
as a lattice(see Fig. 1). The numbers of column and row of
this system are assumed as “N” respectively. We name each
oscillator OSC(k,l)(0 ≤ k and l ≤ N − 1). An actual circuit
is made as N = 5(see Fig. 2). A voltage of each oscillator isnamed v(k,l) , and a current of a inductor in each oscillator isnamed 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 a coupling parameter.
The ε corresponds to a nonlinearity of each oscillator. This
circuit is simulated by using the fourth order 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)dτ = y(0,a) ,(3)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)dτ = y(N,a) ,(4)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)- 463 -− 4x(k,l) ) + ε(y(k,l) − 13 y3(k,l) ).

(0,0)

(0,1)

L0

OSC

L0
(1,0)

L0

L0

OSC

(1,1)

OSC

OSC

L0

B

i(k,l)

L0
L0

OSC

L C

v(k,l)

L0

OSC

L0

A

(0,N-1)

OSC

L0

L0
R1

L0
(N-1,0)

L0
L0

OSC

OSC

+
-

L0

L0
OSC

(N-1,N-1)

R2
R3

Figure 1: Circuit model.

Figure 3: Computer simulation result of the phase-inversion
waves in in-and-anti-phase synchronization for vertical direction and in-phase synchronization for horizontal direction(N = 10, α=0.010, and ε=0.050).











Figure 2: Photo of actual circuit.

dx(a,l)dτ = y(a,l) ,(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
A computer simulation result of existing the phaseinversion waves in in-and-anti-phase synchronization for vertical direction and in-phase synchronization for horizontal direction are shown in Fig. 3 when N = 10. In-and-anti-phasesynchronization is a synchronization phenomena which thein-phase synchronization and the anti-phase synchronization
are alternately existing. The Fig. 3-A expresses an attractor of each oscillator(current vs. voltage). The Fig. 3-B expresses itinerancy of phase diﬀerence by which sum of voltages of adjacent 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. The signs of the initial values of the voltages areshown in Fig. 4. In Fig. 3, we can observe phase-inversion
waves which simultaneously propagate to horizontal direction in in-phase synchronization and to vertical direction in
in-and-anti-phase synchronization.
4. Propagation velocity
We analyze a propagation velocity by using numerical experiments. The propagation velocity is calculated by using
time of which the phase-inversion wave propagates from the
1st row to the Nth row or from the 1st column to the Nth column. Observation conditions are fixed as follows:
1. N is fixed as 10.

13.00e-003

Propagation velocity[osc/τ](top:(a, b)=(0, 1),bottom:(a, b)=(N−1, N−2),both:0 < l < N−1.)

Propagation velocity[osc/τ]Figure 4: Sign of initial value of each oscillator of Fig. 3.

[Edge]

12.0e-003
11.0e-003
10.0e-003
9.00e-003
8.00e-003
7.00e-003
6.00e-003
5.00e-003
4.00e-003
3.00e-003

2.00e-003

0

0.005

α

0.010

0.015

0.02

13.00e-003
12.0e-003
11.0e-003
10.0e-003
9.00e-003
8.00e-003
7.00e-003
6.00e-003
5.00e-003
4.00e-003
3.00e-003

2.00e-003

0

0.005

α

0.010

0.015

0.02

(a) Vertical phase-inversion wave. (b) Horizantal phase-inversion wave.

Figure 5: Propagation velocities of the phase-inversion wave.
2. The nonlinearity is fixed 0.080, and the coupling parameter is changed form 0.005 to 0.020 every 0.005.
3. Time of that the phase-inversion wave propagate from
OSC(1,0) to OSC(1,4) or from OSC(0,1) to OSC(4,1)
is investigated because two phase-inversion waves exist
in each column or in each row.
The propagation velocity of phase-inversion wave is shownin Fig. 5. The vertical axes are the propagation velocity. The
horizontal axes are the coupling parameter α. The propagation velocity is calculated by Eq. (8).
M
(8)
propagation velocity = ,Twhere the number of oscillator propagating the phaseinvertion wave is assumed as “M,” and a propagation time isexpressed as “T .” Comparison of each propagation velocity
is shown in Table 1. We can see that the propagation velocity
increases with the increasing of the coupling parameter. The
velocity of the phase-inversion wave in the in-phase synchronization is slower than the velocity of the phase-inversion
wave in the in-and-anti-phase synchronization.
5. Mechanisms
We can observe a phenomenon that a vertical phaseinversion wave and a horizontal phase-inversion wave reflect at a corner. Vertical phase-inversion waves propagate in- 464 -(k, l)(a)V

Table 1: Comparisons of the propagation velocities.
OSC(k, l)vertical in the in-and-anti-

horizontal in the in-and-anti-

phase synchronization

phase synchronization

0.005

2.617×10−3

2.498×10−3

0.010

5.511×10−3

4.969×10−3

0.015

8.777×10−3

7.424×10−3

0.020

1.220×10−2

9.917×10−3

OSC(k+1, l)a+1

(k+1, l)(a)
a-1

a

a+1

Time

Figure 6: The detection method of the instantaneous frequencies and the phase diﬀerences.
faaa

(9)

6
f(3,0)fiia

2

4

f(0,0)

f(2,0)

Phase Difference [rad]32(0,0)(1,0)



-

5

6

0

5
(0,0)(0,1) andand(1,0)(1,1)(2,0)(2,1)1
6(2,0)(3,0)(1,0)(2,0)-2Time[ ]
Phase difference(a) Vertical phase-invertion wave.

Instantaneous Frequency

f aaa

f(0,2)

8f(0,3)f iaa

6
f(0,1)

2

4

f(0,0)

f iia

Time[ ]
Instantaneous frequency

Phase Difference [rad]3

Φ(k,l)(k+1,l) (a) =(10)

Reflection mechanism at a corner is shown in Table 2(seeFig. 7). In upside of Figs. 7(a) and (b), the vertical axes are
instantaneous frequency, and the horizontal axes are time. In
downside of Figs. 7(a) and (b), the vertical axes are the phase
diﬀerence, and the horizontal axes are time.
6. Experimental results
The actual circuit experimental result of N = 5 is shownin Fig. 8. The numerical experimental result is shown inf(1,0)

fiaa

Time[ ]
Instantaneous frequency

where “a” expresses the number of times of the voltage positive peak value, namely τ(k,l) (a) means the time of OSC(k,l)
as shown in Fig. 6. Similarly, τ(k+1,l) (a) and τ(k,l+1) (a) aredefined. Some frequencies are observed in this phenomena.
These frequencies are needed to consider the synchronization
states for the vertical direction and horizontal direction, because this system is 2-dimensional array. Three phase states
have to be considered for an oscillator in edges of 2D lattice,because the in-phase synchronization and the anti-phase synchronization are exist in same time. Therefore, three types
synchronization states are observed in edges of the lattice asfollows:
1. Two of three phase states are the in-phase synchronization state. Another phase state is the anti-phase synchronization state.
2. Two of three phase states are the anti-phase synchronization state. Another phase state is the in-phase synchronization state.
3. Three synchronization states are the anti-phase synchronization.
The 1st type is called fiia . The 2nd type is called fiaa . The
3rd type is called faaa .
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)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. (10) (see Fig. 6).
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)× 2π [rad].
Φ(k,l)(k,l+1) (a) =τ(k,l) (a) − τ(k,l) (a − 1)

a

Time

the in-and-anti-phase synchronization and horizontal phaseinversion waves propagate in the in-phase synchronization.
This mechanism is analyzed by using an instantaneous frequency of each oscillator and phase diﬀerences between adjacent oscillators. The coupling parameter is fixed as α=0.01,
and nonlinearity is fixed as ε=0.15.
The instantaneous frequency is defined as Eq. (9).
1,f(k,l) (a) =τ(k,l) (a) − τ(k,l) (a − 1)a-1

VInstantaneous Frequencyα2(0,0)(0,1)(0,0)(1,0) andand(0,1)(1,1)(0,2)(1,2)

57

(0,2)(0,3)(0,1)(0,2)

0

1 3Time[ ]
Phase difference(b) Horizontal phase-invertion wave.

Figure 7: Transitions of instantaneous frequencies and phase
diﬀerences by the reflection of two phase-inversion waves at
a corner.
Fig. 9. We can observe phase-inversion waves which simultaneously propagate to horizontal direction and vertical direction in the in-and-anti-phase synchronization. The- 465 -

Table 2: Reflection mechanism of two phase-inversion waves at a corner (see Fig. 7).
no.

Vertical phase-inversion wave

0

At first, the instantaneous frequency f(1,0) is fiia , because
Φ(0,0)(1,0) is π, and Φ(1,0)(2,0) and Φ(1,0)(1,1) are 0. The verti-no.

Horizontal phase-inversion wave

0

At first, the instantaneous frequency f(0,1) is fiia , because
Φ(0,1)(1,1) is π and Φ(0,0)(0,1) and Φ(0,1)(0,2) are 0. The horizontal

cal phase-inversion wave comes from 12th row to 2nd row in
the in-and-anti-phase synchronization in each column.
1Φ(1,0)(2,0) starts to change from 0 toward −π by the verticalphase-inversion wave.

phase-inversion wave comes from 11th column to 2nd column
in the in-phase synchronization in each row.
1Φ(0,1)(0,2) starts to change from 0 toward π by the horizontal phase-inversion wave. f(0,2) is changing to fiaa , because
Φ(0,1)(0,2) is 0, Φ(0,2)(1,2) is π and Φ(0,2)(0,3) is changing to π.

23f(1,0) starts to change from fiia to fiaa , because Φ(1,0)(2,0) ischanging from 0 to −π, Φ(0,0)(1,0) is π and Φ(1,0)(1,1) is 0.
Φ(0,0)(1,0) starts to change from π toward 2π, because f(1,0) ischanging from fiia to fiaa .

2
3

f(0,1) starts to change from fiia toward fiaa , because Φ(0,1)(0,2)
is changing from 0 to π ,Φ(0,0)(0,1) is 0 and Φ(0,1)(1,1) is π.
Φ(0,0)(0,1) starts to change from 0 toward π, because f(0,1) ischanging from fiia to fiaa .

4

f(0,0) almost never changes, because Φ(0,0)(0,1) is changing from 0 to π and Φ(0,0)(1,0) is changing from π to 2π.

5Φ(0,0)(1,0) continuously changes to 3π, because the in-phase synchronization can’t exist at an edge in the in-and-anti-phase synchronization.
Φ(0,0)(0,1) continuously changes to 2π, because the anti-phase synchronization can’t exist at an edge in the in-phase synchronization.

6

f(1,0) starts to change toward faaa , because Φ(0,0)(1,0) is changing to 3π and Φ(1,0)(1,1) is almost π yet. Φ(1,0)(2,0) is start tochange from −π to −2π, because f(1,0) is changing toward faaa
and a diﬀerence between f(1,0) and f(2,0) is increased. f(2,0)

6
7

Φ(0,1)(0,2) starts to change to 2π, because f(0,1) is lower thanf(0,2) . f(0,1) starts to change to fiia , because Φ(0,0)(0,1) is changing to 2π, Φ(0,1)(1,1) is changing to 3π and Φ(0,1)(0,2) is changing
to 2π.

starts to change toward fiia , because Φ(1,0)(2,0) is changing to−2π, Φ(2,0)(2,1) is almost π yet and Φ(2,0)(3,0) is 2π.
f(1,0) starts to change toward fiia , because Φ(0,0)(1,0) changes to
3π, Φ(1,0)(1,1) changes to 2π and Φ(1,0)(2,0) is changing to −2π.

f(0,2) is higher than f(0,1) , because Φ(0,1)(0,2) and Φ(0,2)(0,3) is πand Φ(0,2)(1,2) is changing to 3π.

8f(0,2) can’t arrive at faaa and starts to change to fiaa , because
Φ(0,1)(0,2) is changing to 2π. Because f(0,2) is changing to fiaa ,f(0,2) is lower than f(0,3) and Φ(0,2)(0,3) starts to change to 2π.
f(0,2) continuously changes to fiia , because Φ(0,1)(0,2) is changing to 2π. Φ(0,2)(1,2) is 3π and Φ(0,2)(0,3) is changing to 2π.

The phase-inversion wave reflects at a corner by this mechanism.

Figure 8: Circuit experimental result(N = 5, R1  R2  5.1
kΩ, R3 1.6kΩ, C 5.6nF, L 10mH and L0  20mH).
propagation velocity of actual
circuit is normalized for time√
(propagationvelocity × L0C) because the velocity has tobe compared with simulation result. When L0 is 30mH, α isaround 0.33. When L0 equals 20mH, α is 0.50. The normalized propagation velocity of L0 = 30mH is 0.92 times of thenormalized propagation velocity of L0 = 20mH. In simulation results, the propagation velocity of α = 0.5 is 0.90 timesof the propagation velocity of α = 0.33. We can observed
almost same values between simulation results and experimental results. We can think that our simulation results canbe observed in actual experiments.
7. Conclusion
We clarified that phase-inversion waves simultaneously
propagate to horizontal direction and vertical direction in inand-anti-phase synchronization by using actual circuit experiments and numerical experiments. To increase of propaga-

Figure 9: Computer simulation result(N = 5, α = 0.50 and
ε = 0.80).
tion velocity was made clear when the coupling parameters
are increased. Furthermore, the reflection mechanism of thephase-inversion waves was analyzed at a corner.
Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 25730152.
References[1] C. M. Gray, “Synchronous Oscillations in Neuronal Systems: Mechanisms and Functions,” Journal of Computational Neuroscience 1, pp. 11–38, 1994.
[2] How-Foo Chen; Jia-Ming Liu; “Unidirectionally Coupled Synchronization of Optically Injected Semiconductor Lasers,” IEEE Journal of Selected Topics in Quantum
Electronics, vol. 10, issue 5, pp.918-926, 2004.
[3] S. YANANE, M. YANAUCHI, “Propagation and Reflection of Phase Diﬀerences on a Lattice of Coupled Oscillators,” Proc. ISCAS’09, pp.1895-1898, May 2009.

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