ECCTD’01 - European Conference on Circuit Theory and Design, August 28-31, 2001, Espoo, Finland

Phase-Inversion-Waves in Coupled Oscillators
Synchronizing at In-and-Anti-Phase
Masahiro OKUDA∗, Masayuki YAMAUCHI∗, Yoshifumi NISHIO∗ and Akio USHIDA∗
Abstract - Recently, we have discovered wave
propagation phenomena; which are continuously
existing waves of changing phase states between
two adjacent oscillators from in-phase to antiphase or from anti-phase to in-phase in van derPol oscillators coupled by inductors as a ladder.
We named the phenomena as “phase-inversionwaves.”
In this study, phase-inversion-waves
which exist in the state of in-and-anti-phase synchronization have been found. We observe thephenomena by circuit experiments and computer
calculations, and investigate them.

Figure 1: Coupled van der Pol oscillators as a ladder.

1. Introduction
A lot of studies on synchronization phenomenaof coupled oscillators have been carried out up to
now [1]-[13]. Endo et al. have reported a details oftheoretical analysis and circuit experiments aboutsome coupled oscillators as a ladder, ring and twodimensional array [2]-[4].
Recently, the authors have discovered wave propagation phenomena that phase states between adjacent oscillators change from in-phase to anti-phase orfrom anti-phase to in-phase in oscillators coupled byinductors as a ladder [12]. We named the phenomena as “ phase-inversion-waves.” It is very important to analyze the phase-inversion-waves and makeclear the mechanism of the generation, because it
is similar to the propagation phenomena of electrical information in axial ﬁber of nervous system. In[12], we explained the mechanism of the generationof the phase-inversion-waves by using the relationship between phase states and oscillation frequencies. And we conﬁrmed that the phenomena could beexplained by a simpliﬁed mathematical model [13].
In the circuit, we can observe a synchronization
phenomenon in which in-phase and anti-phase exist
alternately. In this study, we name the synchronization state as “in-and-anti-phase synchronization.”
This is similar to the phenomenon reported in [11].
In this study, we observe phase-inversion-waves inthe state of in-and-anti-phase synchronization. We
carry out circuit experiments and computer calculations and investigate the phenomena.

2. Circuit Model and Phase-InversionWaves
Circuit model is shown in Fig. 1. N van der Pol∗ Department of Electrical and Electronic Engineering,
Tokushima University, 2–1 Minami-Josanjima, Tokushima,
770-8506 Japan. E-mail: okuda@ee.tokushima-u.ac.jp

oscillators are coupled by coupling inductors L0 . In
the computer calculations, we assume the v − i characteristics of nonlinear negative resistors in the circuit by the following functions.
ir (vk ) = −g1 vk + g3 vk3(g1 , g3 > 0)(1)

The circuit equations governing the circuit in Fig. 1
are expressed as ẋk = yk
1 3(2)yẏ=−x+α(x−2x+x)+εy− kkk+1
kk−1k3 k(k = 1, 2, · · · , N )wherex0 = x1 , xN +1 = xN ,
iL1 k =d= “ · ”,dτ

Cg1xk , vk =3L1 g3

g1yk ,3g3


L1, ε = g1t = L1 Cτ, α =L0

L1.
C(3)

It should be noted that α corresponds to the coupling and that ε corresponds to the nonlinearity.
Equation (2) are calculated by using the fourth-order
Runge-Kutta method.
Figure 2 shows typical examples of the phaseinversion-waves reported previously. They are computer calculated results from the circuit with thesize of N = 30. Vertical axes are sum of two voltages of adjacent oscillators and horizontal axes aretime. White regions in the diagram correspond to
the states that sum of voltages of the two oscillators are close to zero, namely adjacent two oscillators synchronized at anti-phase. While, black regions correspond to the states that sum of voltages

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(a)(a) Extinction by collision of two waves.

(b)

Figure 3: Example of in-and-anti-phase synchronization. (a) Circuit experimental result, L0 =1170mH,
L1 =200mH, C=68nF, r=1kΩ. (b) Computer calculated result, α = 0.10, ε = 0.30, ∆τ = 0.001.
synchronization state can be observed only when N
is even number. While the phase-inversion-waves areobserved when N is both even number and odd number. And, the oscillation frequency for the in-andanti-phase synchronization fmid is almost average ofthe oscillation frequency for the in-phase synchronization flow and the oscillation frequency for theanti-phase synchronization fhigh .

(b) Reﬂection by collision of two waves.
Figure 2: Examples of phase-inversion-waves. α =0.10, ε = 0.30, ∆τ = 0.001.

of the two oscillators oscillates with large amplitude,
namely adjacent two oscillators synchronized at inphase. In the Figures, we can see the wave propagation, reﬂection and extinction.
It has been known that two van der Pol oscillators
coupled by inductor have two stable phase states,
namely in-phase synchronization and anti-phase synchronization. Further, the oscillation frequency forthe in-phase synchronization is smaller than the oscillation frequency for the anti-phase synchronization. By using these features, we could explain themechanism by using the relationship between phasestates and oscillation frequencies. (See [12] for thedetails.)

3.2 Experimental Results
Figures 4 and 5 show examples of phase-inversionwaves in the state of in-and-anti-phase synchronization. Figure 4 is the array with the size of N ≤ 9 bycircuit experiments and computer calculations.
In Fig. 4(a), phase-inversion-waves do not exist.
A phase-inversion-wave is existing continuously inFig. 4(b). The change of phase states from in-phaseto anti-phase or from anti-phase to in-phase is propagated. In Fig. 4(c), two waves are existing and awave is propagating on the heels of another wave.
Figure 5 shows computer calculated results obtained for the case of N = 29 and 30. We can observe
phase-inversion-waves similar to Fig. 4.
On the system, we could not observe wave extinction by collision of two waves like Fig. 2(a).

3.3 Single Phase-Inversion-Wave
3. Phase-Inversion-Waves in the Stateof In-and-Anti-Phase Synchronization
3.1 In-and-Anti-Phasetion

Synchroniza-

In phase-inversion-waves reported previously [12],single wave could not exist. However, single wave
can exist on the system of coupled oscillators of odd
number. (See Fig. 4(b).) Figure 6 shows outline
chart of single wave generated at the edge of thearray with the size of N = 9. Generation of thewave is explained as follows:

In the circuit model, we can observe another typeof synchronization as shown in Fig. 3. Vertical axes
are the amplitudes of voltages and horizontal axes
are time. We name this as “in-and-anti-phase synchronization” because in-phase and anti-phase areexisting alternately. Also, in this synchronization
state, the edge of the array and its adjacent oscillator can not synchronize at in-phase. Hence, thisIII-206

1. Set initial conditions that in-phase and antiphase exist alternately, OSC1 and OSC2 synchronize at anti-phase, OSC8 and OSC9 synchronize at in-phase. ( 1 in Fig. 6.)
2. A phase-inversion-wave is generated because
phase state between OSC8 and OSC9 changes
from in-phase to anti-phase. ( 2 in Fig. 6.)

Circuit experiments

Computer calculations(a) No wave (N = 6).

(a) Single wave (N = 29).

(b) Single wave (N = 9).

(b) Double waves (N = 30).
(c) Double waves (N = 8).
Figure 4: Experimental results.
Circuit experiments, L0 = 1170mH, L1 = 200mH, C = 68nF,
r = 1kΩ. Computer calculations, α = 0.10, ε =0.30, ∆τ = 0.01.

Figure 5: Examples of phase-inversion-waves. α =0.10, ε = 0.30, ∆τ = 0.01.

3. Phase state between OSC7 and OSC8 starts tochange from anti-phase to in-phase. ( 3 in Fig.
6.)
Figure 6: Single wave (outline chart).
4. Phase state between OSC6 and OSC7 changes
from in-phase to anti-phase, wave is propagatedtoward OSC1 . ( 
4 in Fig. 6.) Single wave
exists because OSC8 and OSC9 synchronize atanti-phase.
The single wave has been observed for the ﬁrst
time in this study.

where τk (n) is time when the voltage of OSCk crosses
0[V] at n-th time.
Mechanism of wave propagation can be explained
as follows:

3.4 Mechanism of Wave Propagation
Figure 7 shows phase diﬀerences and oscillation
frequencies of OSC4 ∼ OSC6 . Single wave is propagated in the ﬁgure. It should be noted that Φk,k+1
is phase diﬀerence between OSCk and OSCk+1 andthat fk is oscillation frequency of OSCk . We deﬁnethem as follows:τk (n) − τk+1 (n)×πτk (n) − τk (n − 1)

(4)12(τk (n) − τk (n − 1))(5)

Φk,k+1 =fk (n) =III-207

1. Let us assume that the circuit synchronizes atin-and-anti-phase and that single wave is goingto reach OSC6 from the direction of OSCN .
2. As the phase state between OSC6 and OSC7 approaches anti-phase from in-phase, f6 changes
from fmid to fhigh . Hence, Φ5,6 starts tochange. ( 1 in Fig. 7.)
3. As the phase state between OSC5 and OSC6approaches in-phase from anti-phase, f5 starts
to change from fmid to flow . ( 2 in Fig. 7.)
4. As f5 changes from fmid to flow , the phase statebetween OSC4 and OSC5 approaches anti-phasefrom in-phase. (2 in Fig. 7.) Before f6 reaches
fhigh , it approaches fmid by inversion of thephase states between adjacent oscillators.

phase states and oscillation frequencies.

References[1] D.A. Linkens “Analytical Solution of Large
Numbers of Mutually Coupled Nearly Sinusoidal Oscillators” IEEE Trans. Circuits Syst.,
vol. cas-21, no. 2, pp. 294-300, Mar. 1974.
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Syst., vol. 23, no. 2, pp. 100-113, Feb. 1976.
[3] T. Endo and S. Mori “Mode Analysis of TwoDimensional Low-Pass Multimode Oscillator”
IEEE Trans. Circuits Syst., vol. 23, no. 9, pp.
517-530, Sep. 1976.

(a) Phase diﬀerence.

[4] T. Endo and S. Mori “Mode Analysis of a Ringof a Large Number of Mutually Coupled vander Pol Oscillators” IEEE Trans. Circuits Syst.,
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(b) Oscillation frequency.
Figure 7: Wave propagation (computer calculatedresults).

[6] L. Fabiny and K. Weisenfeld “Clustering Behavior of Oscillator Arrays” Physical Review A,
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5. Before f5 reaches flow , it approaches fmid byinversion of the phase states between adjacentoscillators. f5 is equal to fmid when the phasestates invert almost. ( 3 in Fig. 7.)
After the wave reﬂects at the edge of the array,
when the wave propagated to the direction of OSCN
reaches OSC4 , the phase states and oscillation frequencies change in a similar manner. (4 in Fig. 7.)
Although the above explanation is for the case ofwave propagation, wave reﬂection can be explainedin a similar manner.

4. Conclusions
In this study, we observed phase-inversion-waves
in the ladder of coupled oscillators synchronizing at
in-and-anti-phase, in which in-phase and anti-phaseare existing alternately. We carried out circuit experiments and computer calculations.
In the previous study, wave propagation, reﬂection at the edge of array, reﬂection by collision of
two waves and extinction by collision of two waves
could be observed. However, extinction could not
be observed in this study. Further, single phaseinversion-wave could be observed for the ﬁrst time.
In phase-inversion-waves in the state of in-andanti-phase synchronization, oscillation frequencieschange around single stable frequency fmid . By using this feature, we could explain the mechanism ofwave propagation by using the relationship between[8] S. Moro, Y. Nishio and S. Mori “Synchronization Phenomena in Oscillators Coupled by
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E78-A, no. 2, pp. 244-253, Feb. 1995.
[9] S. Moro, Y. Nishio and S. Mori “On Coupled
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[10] H. Daido “Strange Waves in Coupled-Oscillator
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[11] Y. Setou, Y. Inami, Y. Nishio and A. Ushida“On a Ladder of Oscillators Sharing Inductors”
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[12] M. Yamauchi, M. Wada, Y. Nishio and
A. Ushida “Wave Propagation Phenomena ofPhase States in Oscillators Coupled by Inductors as a Ladder,” IEICE Trans. Fundamentals,
vol. E-82A, no.11, pp. 2592–2598, Nov. 1999.
[13] M. Okuda, M. Yamauchi, Y. Nishio andA.Ushida “Analysis of Wave-Motion in Coupled Oscillators by Simpliﬁed Model” Proc. ofNDES2000, pp. 105-108, May. 2000.

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