2016 International Symposium on Nonlinear Theory and Its Applications,NOLTA2016, Yugawara, Japan, November 27th-30th, 2016

Investigation of Attracting Force to Synchronization States on Coupled
Oscillator System by Using Electric Power
Ryoji Fukumasa† , Masayuki Yamauchi† and Yoshifumi Nishio‡†Hiroshima Institute of Technology
2-1-1, Miyake, Saeki-ku, Hiroshima, 731–5193, Japan
Phone/FAX:+81-82-921-4248
E-mail: ma16009@cc.it-hiroshima.ac.jp‡Tokushima University
2-1, Minamijyosanjima-cho, Tokushima, 770–8506, Japan
E-mail: nishio@ee.tokushima-u.ac.jp

Abstract
In this study, the attracting forces to a steady states aretheoretically analyzed by using reactive powers in coupledoscillator system, and are obtained by using a simulation.
The attracting forces are investigated in changing a coupling parameter by using our theoretical method.
1. Introduction
We can observe synchronization phenomena in our body.
There are a synchronization of among pacemaker cells in
our heart, and the pulse synchronization phenomenon canbe observed in other nervus system, too. Therefore, we cansay that the synchronization phenomenon is one of mostimportant phenomena in this world. The synchronization
phenomena can be observed on electric circuits. Manykinds of synchronization phenomena and characteristics ofthe phenomena were reported by many researchers[2]. Thesynchronization phenomena have been analyzed by manymethods. The averaging method is often used when thesynchronization phenomena are theoretically analyzed[2].
The method is hard to be used for transient state.
In this study, an attracting force, which two oscillators
are attracted to a synchronization state, are investigated byusing reactive powers of the system and simulations. Furthermore, the attracting force is investigated in changing acoupling parameter by using our theoretical method.
2. Circuit Model
Our circuit model is shown in Fig. 1. The van der Pol
oscillators are coupled by inductor LC . An inductor and acapacitor of each van der Pol oscillator are shown as “L”
and “C” respectively. Characteristic of nonlinear negative
resistor of k-th oscillator is shown as f (vk ) in Eq. (1).
f (vk ) = −g1 vk + g3 v3k(k = 1 or 2)(1)

Circuit equations of this circuit are normalized by using Eq.
(2). The normalized equations are shown in Eq. (3).
√√√
Cg1
g1
i=xk , v =yk , t = LCτ,3Lg3
3g3
Lα = , ε = g1
Lc

√

g2
L, δ= 1 .
C3g3

(2)

Figure 1: Circuit model.
dxk= yk ,dτdyk
1= −xk + α(xa − 2xk + xb ) + ε(yk − y3k ).
dτ3(3)(If k = 1, a = N and b = 2.
If k = N, a = N − 1 and b = 1.
If 2 ≤ k ≤ N − 1, a = k − 1 and b = k + 1.)
Where α expresses a coupling parameter and ε showsnonlinearity of each oscillator.
2.1. Calculation of reactive power
An instantaneous electric power of each oscillators andan instantaneous electric power of each inductor between
adjacent oscillators are calculated by which each oscillation wave shape is assumed as a sinusoidal wave. Eachcurrent xk (k = 1 or 2) and ezch voltage yk are assumed as
Eq. (4) Amplitudes of two oscillators are assumed as samevalue. Angular frequencies of two oscillators are assumed
as same value, too.
xk = Xsin(ωτ + θk )yk = ωXsin(ωτ + θk )(4)<Instantaneous electric power of the inductor in eachoscillator>
δ(5)PLk = xk ykε<Instantaneous electric power of the capacitor in eachoscillator>
δ dyk
(6)
PCk = ykε dτ<Instantaneous electric power of the coupling inductor>
PLc(2,1) =- 385 -αδ(y1 − y2 )(x1 − x2 )ε(7)A normalized equation of a total instantaneous reactive
power are assumed as Eq. (8).
)
2 (∑δ dykδ
xk yk + yk
Prall =εε dτ
k=1(8)αδ+ (y1 − y2 )(x1 − x2 )ε2.2. Derivation of angular frequency
In this system, we can consider that a power eﬀect isbest when a reactive power, which can be assumed as sumof instantaneous electric powers of L, C and LC , is zero.
In other words, we can guess that this system is a steadystate when the reactive power is zero. In this study, angular frequencies of the in-phase synchronizations and theanti-phase synchronizations are obtained when the reactive
powers are zero.
Prall = 0(9)We want to calculate Eq. (8), but we can’t calculate theequation without an oscillation angular frequency ω and anamplitude X . However, when θ1 and θ2 are zero or θ1 is 0and θ2 is π, ω can be obtained by using Eq. (9).
<Phase angles of the in-phase synchronization>(θk = 0(k = 1 or 2))(10)<Phase angles of the anti-phase synchronization>()
θ1 = 0(11)θ2 = πFirstly, an in-phase synchronization frequency ωin andan anti-phase synchronization frequency ωanti of this circuit are calculated by using the Eq. (9). A phase of eachoscillator is set as θk .
<Phase angles of the in-phase synchronization>(θk = 0(k = 1 or 2))(12)<Phase angles of the anti-phase synchronization>()
θ1 = 0(13)θ2 = πThe angular frequency of the in-phase synchronization isachieved by which Eq. (12) is applied to Eq. (9).
<Anglular frequency of the in-phase synchronization>
ωin = 1(14)The angular frequency of the anti-phase synchronization isachieved by which Eq. (13) is applied to Eq. (9).
<Anglular frequency of the anti-phase synchronization>√(15)
ωanti = 1 + 2α
We assume that each angular frequency ω of each phase
diﬀerence (θ2 − θ1 )[degree] exists between ωin and ωanti
and linearly vary between ωin and ωanti . Therefore, ω iscalculated by the following equation.
ω=ωanti − ωin× (θ2 − θ1 ) + ωin
180

(16)

Next, amplitude X is calculated. A total active power ofthis circuit is obtained as a sum of powers of a nonlinear
negative resistor in each oscillator(see Eq. (17)).
Paall =2∑
k=1(−δd2 xk δ d4 xk+)
3 dτ4dτ2(17)An amplitude X of each oscillator is calculated, when
Paall is integrated in a period and the result is assumed zero
as follows.
∫ τPaall dτ = 00(18)2X=ω
We calculate Prall of each phase diﬀerence by using theω and the X.
3. Attracting Force
The attracting force is considered by using reactive powers and simulations.
3.1. Analyzing method by using reactive powers
We set θ = θ2 − θ1 . Waveforms of reactive powers areshown in Figs. (2)–(6) in changing the phase diﬀerence θ.
We can understand that the amplitude is zero when the θis 0 degrees or 180 degrees. In other words, if a conditionof the system is a steady state, the amplitude of the reactive power is zero, and if the condition is not the steadystate, the amplitude is not zero. Therefore, we think thatthe system becomes unstable when a value, which the reactive power is squared and integrated in a period, becomelarge. The value is expressed as PrT .
∫ τ
PrT =Prall2 dτ(19)0The PrT is calculated in changing phase diﬀerence θ andshown in Figs. (7) and (8). The Fig. (7) shows results ofwhen the α is set as 0.05 and the ε is 0.1, and the Fig. (8)shows results of when the α is set as 0.1 and the ε is 0.1, We
can understand that PrT becomes a maximum value when
θ = 90 degrees. In under 90 degrees, when the θ becomessmall, PrT becomes small. In over 90 degrees, when theθ becomes large, PrT becomes small. We assume that θchanges so that the PrT becomes small. In other words, theθ is attracted to zero under 90 degrees, and θ is attracted to180 degrees over 90 degrees. If change of the PrT is large
when the θ is a little changed, we think that the attracting
force is large. Therefore, we calculate gradient values of
the graph of the PrT . The gradient values are multiplied by-1, because we want to show the results as negative values
in a domain of which the θ decreases to zero, and as positive
values in a domain of which the θ increases to 180 degrees.
The inversion gradient values g are shown in Figs. (9) and(10)(see Eq. (20)).

- 386 -PrT

1202ππ0

Figure 2: Instantaneous reactive power Prall(θ=0 degrees).

Figure 7: Theoretical results of PrT (α = 0.05 and ε =0.10).
5002πPrTπ0Figure 3: Instantaneous reactive power Prall(θ=45 degrees).

Figure 8: Theoretical results of PrT (α=0.10 ε=0.10).

0πInversion gradient value of PrT0.5

2π0090180-0.5

Figure 4: Instantaneous reactive power Prall(θ=90 degrees).

Figure 9: Theoretical results of g(α=0.05 ε=0.10).

0πInversion gradient value of PrT2.02π0090180-2.0

Figure 5: Instantaneous reactive power Prall(θ=135 degrees).

Figure 10: Theoretical results of g(α=0.10 ε=0.10).
dPrT(20)dθ
We can think that the attracting force to stable state isstrongest when an absolute value of the g is a maximumvalue. Therefore, we can understand that when θ=45 degrees and 135 degrees attracting force become strongest.
g=−0π

2π

3.2. Analyzing method by using simulations
Figure 6: Instantaneous reactive power Prall(θ=180 degrees).

An attracting force of each θ is investigated by using asimulator. An initial phase diﬀerence between two oscillators is changed by which initial values of a voltage and- 387 -a current of each oscillator are changed. A calculation
method of phase diﬀerence is shown as follows(see Fig.
11). In the Fig. 11, two sinusoidal waves show voltagesof two oscillators. The a1 shows the first positive peak of aoscillation waveform of an oscillator after τ = 0, and the a2
expresses the second positive peak. The b1 shows the first
positive peak of a oscillation waveform of another oscillator. Time of a1 , a2 and b2 are expressed τa1 , τa2 and τb2 .
The θ is calculated by using Eq. (21).
τb2 − τa2
θ=× 360[degree](21)
τa2 − τa1
In this paper, we investigate how much the θ is changed
during 1τ for each the initial phase diﬀerence. A changing
value of θ is shown as ∆θ and calculated by Eq. (22).
()τb3 − τa3 τb2 − τa21∆θ =−×× 360[degree]τa3 − τa2 τa2 − τa1
τa3 − τa2
(22)
The simulation results are shown in Figs. 12 and 13. Thehorizontal axes show the θ, and the vertical axes show the∆θ in Figs. 12 and 13. Therefore, this graph means anattracting force g which is strong when a absolute value of∆θ is large value.
We can observe that the shapes of graphs of simulation
results are same shape with theoretical graphs, basically.
4. Comparison

0.15 or 0.20. The theoretical results are shown in Fig. 14.
We can understand that the attracting force g becomes large
as the coupling parameter α becomes large.
5. Conclusions
In this study, the attracting forces to a steady states weretheoretically analyzed by using reactive powers and wereobtained by a simulation in a coupled oscillator system.
The theoretical results and the simulation results were observed same results, basically. Furthermore, we investigated that the attracting force becomes strong as coupling
parameter is increased by using our theoretical method.
Acknowledgment
This work was supported by JSPS KAKENHI Grant
Number 25730152.
References[1] Masayuki Yamauchi, Masahiro Wada, Yoshifumi
Nishio and Akio Ushida, “Wave Propagation Phenomena of Phase States in Oscillators Coupled by Inductors
as a Ladder,” IEICE Trans. Fundamentals, vol.E82-A,no.11, pp.2592-2598, Nov. 1999.
[2] Tetsuro Endo and Sinsaku Mori, “Mode Analysis of aRing of a Large Number of Mutually coupled van der
Pol Oscillators,” IEEE Trans. on Circuits And Systems,
vol. cas-25, no. 1, pp.7-18, Jan. 1978.

Attracting forces are investigated in changing the coupling parameter α by our using theoretical method. The
nonlinearity ε is fixed as 0.1 and the α is set as 0.05, 0.10,
0.4
a1b1

a2b22π0a34π3ππb305π090

180-0.46000Figure 13: Simulation result of g(α = 0.10 and ε = 0.10).
Inversion gradient value of PrT

Figure 11: A calculation method of a phase diﬀerence.

90
180090180-60Figure 12: Simulation result of g(α = 0.05 and ε = 0.10).

Figure 14: Theoretical results in changing a coupling
parametar from 0.05 to 0.20 every 0.05.

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