2011 International Symposium on Nonlinear Theory and its Applications
NOLTA2011, Kobe, Japan, September 4-7, 2011

Actual Circuit Experiments and Computer Simulationsof Phase-Inversion Waves
in In-and-Anti-Phase Synchronization on 2D Lattice Oscillators
Seiko Kunihiro† , Ryusei Okamoto†,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
Abstract—We have been observing synchronization phenomena on coupled oscillators systems. A special phenomenon was observed on lattice oscillators by using computer simulations. The special phenomenon is a phenomenonchanging phase states between two adjacent oscillators fromin-phase synchronization to anti-phase synchronization orfrom anti-phase synchronization to in-phase synchronizationin steady state. We call the phenomenon phase-inversion
waves. In this paper, we clarify a mechanism of penetration between two phase-inversion waves in in-and-anti-phasesynchronization on the lattice system. Furthermore, we analyze phase-inversion waves on the coupled oscillators systems as a lattice by using actual circuit experiments and computer simulations.

1. Introduction
There are a lot of synchronization phenomena in thisworld. For example, we can observe biological clocks, thesynchronization of many fireflies, synchronization in combustion of candle, and so on. Furthermore, synchronization
phenomena play an important role in biological and physiological systems[1]-[2].
In our previous study, we observed particular synchronization phenomena on coupled oscillators systems by using
van der Pol oscillators which are coupled by inductors as alattice[3]-[4]. We clarified regions which the phase-inversion
waves can be observed in in-and-anti-phase synchronization
when N = 9, and a mechanism of propagation of a phaseinversion wave in in-and-anti-phase synchronization. Thesestudies analyzed by using computer simulations. In other
hand, on a ladder system, we analyzed the phase-inversion
waves by using actual circuit experiments[5].
In this study, we clarify a mechanism of penetration between two phase-inversion waves in in-and-anti-phase synchronization using instantaneous frequency of each oscillator and phase diﬀerences between adjacent oscillators on thelattice system. Furthermore, we analyze the phase-inversion
waves on the coupled oscillators systems as a lattice by using
actual circuit experiments and computer simulations.

(0,0)

L0

OSC

(0,1)
OSC

L0

L0

(0,N-1)OSC

L0i(k,l)OSC

L C

v(k,l)

(1,1)
(1,0)

OSC

OSC

OSC

L0
R1
+
-

L0
(N-1,0)

OSC

OSC

OSC

R2

(N-1,N-1)

R3

Figure 1: Circuit model
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, l ≤ N − 1). A voltage of each
oscillator is named v(k,l) , and a current of a inductor of each
oscillator is named i(k,l) (see Fig. 1).
3. Computer simulations
We observe on the coupled oscillators systems as a latticeby using computer simulations. An equation of the nonlinear
negative resistor is shown as Eq. (1). Circuit equations of
this circuit are normalized by Eq. (2). The normalized circuit
equations are shown as Eqs. (3)–(7).
ir (v(k,l) ) = −g1 v(k,l) + g3 v3(k,l) .

(1)




Cg1
g1
x(k,l) , v(k,l) = 3gy(k,l) ,i(k,l) = 3Lg
3
3

√
d
= “ · ”, α = LL0 , ε = g1 CL .
t = LCτ, dτ(2)[Corner–top] (left:(a, b)=(0, 1). right:(a, b)=(N − 1, N − 2).)dx(0,a)dτ = y(0,a) ,dy(0,a)dτ = −x(0,a) + α(x(0,b) + x(1,a) − 2x(0,a) )(3)

1+ ε(y(0,a) − y3(0,a) ).
3[Corner–bottom] (left:(a, b)=(0, 1). right:(a, b)=(N − 1, N − 2).)- 480 -

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) )(4)+ ε(y(N,a) − 13 y3(N,a) ).

OSC(0,0)A

B

0OSC(0,5)

0
1
2
3
4
5

OSC(5,0)OSC(5,5)Figure 2: Computer simulation result of the phase-inversion
waves in in-and-anti-phase synchronization when N = 6.
[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) ).

1

2

3

4

5

++−−++++−−++−−++−−−−++−−++−−++++−−++

Figure 3: Sign of initial value of each oscillator when theFig. 2 can be observed.
3.1. Penetration mechanism
We can observe a phenomenon that a vertical phaseinversion wave and a horizontal phase-inversion wave penetrate each other in in-and-anti-phase synchronization. Thismechanism is analyzed by using instantaneous frequency ofeach oscillator and phase diﬀerences between adjacent oscillators when N = 19.
An equation of the instantaneous frequency of OSC(k, l)
is obtained as Eq. (8).
f(k,l) (a) =[Edge](top:(a, b)=(0, 1).bottom:(a, b)=(N−1, N−2).both:0 < l < N−1.)1.
τ(k,l) (a) − τ(k,l) (a − 1)

(8)

(7)

The instantaneous frequency is named f(k,l) (a) where “a” expresses the number of times of the positive peak value ofthe voltage. Time of a-th positive peak value of the voltage
of OSC(k, l) is assumed as τ(k,l) (a)(see Fig. 4). Similarly,
τ(k+1,l) (a) and τ(k,l+1) (a) are decided. Three frequencies areobserved in this system. To consider of the synchronizations for the vertical direction and for the horizontal direction are needed, because this system is 2 dimensional array.
The in-phase synchronization and the anti-phase synchronization exist. Therefore, three types of synchronizations areobserved as follows:+ ε(y(k,a) − 13 y3(k,a) ).

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) are the in-phase synchronization.

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) ,dy(k,a)dτ = −x(k,a) + α(x(k−1,a) + x(k+1,a) + x(k,b) − 3x(k,a) )The α corresponds to a coupling parameter. The ε corresponds to a nonlinearity of each oscillator. This circuit issimulated by using the fourth order Runge-Kutta method andEqs. (3)-(7). A computer simulation result of the phaseinversion waves in in-and-anti-phase synchronization when
N = 6 are shown in Fig. 2. We observe laterally-propagatedand longitudinally-propagated phase-inversion waves in inand-anti-phase synchronization. The Fig. 2–A expresses anattractor of each oscillator(current vs. voltage). The Fig. 2–
B expresses itinerancy of phase diﬀerence by which sum of
voltages of adjacent oscillators is shown along the time(sumof voltage vs. time). The coupling parameter α is fixed as
0.05, and nonlinearity ε is fixed as 0.15.
Figure 3 shows the signs of the initial values of the voltages and currents of each oscillator when the Fig. 2 can beobserved. We can observe some characteristics of phaseinversion waves in in-and-anti-phase synchronization. These
characteristics are a propagation, a penetration, a reflection
at an edge, a reflection at a corner and a reflection between
two phase-inversion waves(see Fig. 2). These characteristics
are shown in Table 1.

2. {x (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.} , {y 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 + 1, l), andOSC(k, l)–OSC(k, l + 1) are another phase synchronization state.} , {z 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) are 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- 481 -

Table 1: Characteristics of the phase-inversion waves in 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.

Reflections at an edge

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

Reflections at a corner

When two phase-inversion waves coming from the vertical direction and the horizontal direction arrive at a corner
oscillator at the same time, the phase-inversion waves reflect and propagate to where they came from.

Reflections between two

When two phase-inversion waves coming from the opposite directions arrive to two adjacent oscillator at same time,phase-inversion waves

the phase-inversion waves reflect and propagate to where they came from.

τ(k, l)(a)OSC(k, l)a-1

afanti-antiInstantaneous Frequency

V

a+1

Time

τ(k+1, l)(a)V

OSC(k+1, l)
a-1

a

a+1

f(5,5)

f(7,5)
andf(5,7)
4f in-anti2f(6,5)

f(6,4)

andf(5,6)

andf(4,6)

f(4,5)
andf(5,4)

Timef in-inFigure 4: The detection method of frequencies and the phase
diﬀerences.

Time[ ](a) Instantaneous frequency
360

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. (9)(see Fig. 4).

and(6,5)(7,5)

Phase Difference [degree]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)(5,6)(5,7)Φ(k,l)(k+1,l) (a) =(9)(4,5)(5,5)and(5,4)(5,5)

180

5

1

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

and

(5,5)(6,5)(5,4)(6,4)-180

Penetration mechanism is shown in Table 2(see Fig. 5). In
Fig. 5(a), the vertical axis is instantaneous frequency, andhorizontal axis is time. In Fig. 5(b), the vertical axis is thephase diﬀerence, and the horizontal axis is time.
4. Actual circuit experiments
We confirm which the phase-inversion waves can be stable
and actually existing in in-and-anti-phase synchronization on
the coupled oscillators systems as a lattice by using actual
circuit experiments. The actual circuit experimental result of
N = 6 shows in Fig. 6. We can observe same phenomenon
by using the computer simulation(See Fig 7). Figures 8 and
9 are extracted from the Figs. 6 and 7. The phase-inversion
waves of horizontal direction exist in in-and-anti-phase synchronization. We can consider that the small diﬀerence between these two results diﬀer from the resistances in real inductors and variations on electric elements.
5. Conclusion
We clarified a mechanism of penetration between twophase-inversion waves in in-and-anti-phase synchronizationon the lattice system. Furthermore, we analyzed of phaseinversion waves on the coupled oscillators systems as a lattice by using actual circuit experiments and computer simulations. We observed same phenomenon in the actual circuit

Time[ ](b) Phase difference

Figure 5: Transitions of phase diﬀerence and frequencies bypenetration between two phase-inversion waves on in-andanti-phase synchronization.
experimental results and the computer simulation results(seeFigs. 6 and 7). We proved that the phase-inversion waves canbe stable and actually existing on actual circuit.
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.
References[1] K. Kotani, K. Takamasu, Y. Ashkenazy, H. E. Stanley,
and Y. Yamamoto “Model for cardio-respiratory synchronization in humans,” Physica E: Stat Nonlin Soft
Matter Phys, 65, 051923, 2002.
[2] Nomura K, Takei Y, Yanagida Y. “Comparison of cardiolocomotor synchronization between running and cycling,” Eur J of Appl Phesiol, 89(3-4):221-229, 2003.
[3] S. Yamane, M. Yamauchi and Y. Nishio, “Classification- 482 -

Table 2: Penetration mechanism of two phase-inversion waves(see Fig. 5).
no.

vertical direction

horizontal direction

0

At this time, Φ(4,5)(5,5) and Φ(6,5)(7,5) are fixed the anti-phase synchronization. Φ(5,5)(6,5) and Φ(5,4)(6,4) are fixed the in-phase synchronization.

At this time, Φ(5,4)(5,5) and Φ(5,6)(5,7) are fixed the anti-phase synchronization. Φ(5,5)(5,6) and Φ(4,5)(4,6) are fixed the in-phase synchronization.

Phase-inversion waves, which changes synchronized state, are arrived at
the number of row is 7 from the number of row is 8.

Phase-inversion waves, which changes synchronized state, are arrived at
the number of column is 7 from the number of column is 8.

A phase diﬀerence Φ(6,5)(7,5) starts to change from the anti-phase syn-

A phase diﬀerence Φ(5,6)(5,7) starts to change from the anti-phase syn-chronization toward the in-phase synchronization by a phase-inversion

chronization toward the in-phase synchronization by a phase-inversion

1

2

3

4

5

wave.

wave.

A instantaneous frequency f(6,5) starts to change from fin−anti toward
fin−in , because Φ(6,5)(7,5) starts to change from the anti-phase synchro-

A instantaneous frequency f(5,6) starts to change from fin−anti toward
fin−in , because Φ(5,6)(5,7) starts to change from the anti-phase synchro-nization toward the in-phase synchronization. f(6,4) also starts to change
from fin−anti toward fin−in , because a phase-inversion wave of adjacentnization toward the in-phase synchronization. f(4,6) also starts to change
from fin−anti toward fin−in , because a phase-inversion wave of adjacentvertical direction arrives at 6th row line at the same time.

horizontal direction arrives at 6th column line at the same time.

Φ(5,5)(6,5) and Φ(5,4)(6,4) start to from the in-phase synchronization to-Φ(5,5)(5,6) and Φ(4,5)(4,6) start to from the in-phase synchronization to-ward the anti-phase synchronization, because f(6,5) and f(6,4) start tochange from fin−anti toward fin−in .

ward the anti-phase synchronization, because f(5,6) and f(4,6) start tochange from fin−anti toward fin−in .

A horizontal phase-inversion wave and a vertical phase-inversion wave arrives at OSC(5,5). f(5,5) starts to change from fin−anti toward fanti−anti .
f(4,5) and f(5,4) start to change from fin−anti toward fanti−anti , because Φ(5,4)(6,4) and Φ(4,5)(4,6) start to change from the in-phase synchronization
toward the anti-phase synchronization.
Φ(4,5)(5,5) starts to change from the anti-phase synchronization to-Φ(5,4)(5,5) starts to change from the anti-phase synchronization to-ward the in-phase synchronization, because f(5,5) starts to change fromfin−anti toward fanti−anti .

ward the in-phase synchronization, because f(5,5) starts to change fromfin−anti toward fanti−anti .

The phase-inversion waves penetrate each other by above mechanism.
horizontal direction

vertical direction

horizontal direction

vertical direction

Figure 6: Actual circuit experimental result of N=6.
(R1  R2 16[kΩ],R3 1[kΩ],C 4.7[nF],L 100[mH] and L0 50[µ H]).

OSC(0,0)-OSC(1,0)

OSC(0,0)-OSC(0,1)

OSC(1,0)-OSC(2,0)

OSC(0,1)-OSC(0,2)

OSC(2,0)-OSC(3,0)

OSC(0,2)-OSC(0,3)

OSC(3,0)-OSC(4,0)

OSC(0,3)-OSC(0,4)

OSC(4,0)-OSC(5,0)

OSC(0,4)-OSC(0,5)

Figure 8: Extraction of the Fig.6 in vertical direction andhorizontal direction.
vertical direction

horizontal direction

OSC(0,0)-OSC(1,0)

OSC(0,0)-OSC(0,1)

OSC(1,0)-OSC(2,0)

OSC(0,1)-OSC(0,2)

OSC(2,0)-OSC(3,0)

OSC(0,2)-OSC(0,3)

OSC(3,0)-OSC(4,0)

OSC(0,3)-OSC(0,4)

OSC(4,0)-OSC(5,0)

OSC(0,4)-OSC(0,5)

Figure 9: Extraction of the Fig.7 in vertical direction andhorizontal direction.

Figure 7: Computer simulation result of N=6.
(α= 0.05 and ε=0.15.)of phenomena on coupled oscillators system as a lattice,”
Proc. NCSP’09, pp.249-252, Mar. 2009.
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Fundamentals, vol.E86-A, no.7, pp.1799-1806, July.
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Propagation Phenomena of Phase States in Oscillators
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