2019 International Symposium on Nonlinear Theory and Its Applications
NOLTA2019, Kuala Lumpur, Malaysia, December 2-6, 2019.

Analysis of Phase-Inversion Waves and Phase-Waves by Instantaneous
Expanding Rates
Rika Hirano† , Masayuki Yamauchi† , and Yoshifumi Nishio‡†Dept. of Elec. and Comp. Eng., Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku, Hiroshima 731-5193, Japan‡Dept. of Elec. and Elec. Eng., Tokushima University, 2-1 MinamiJyosanjima-cho, Tokushima 770-8506, Japan
Email: ma19017@cc.it-hiroshima.ac.jp
Abstract—In this paper, we deﬁne an instantaneous expanding rate which is deﬁned as a change rate of value ofvoltage on Poincare section of which a current is zero ineach oscillator. Furthermore, we deﬁne an quasi Lyapunov
exponent. We observe and analyze the phase-inversion
waves and the phase-waves by using the instantaneous expanding rates and the quasi Lyapunov exponents.
1. Introduction
Various things and living are synchronizing and resonating in this natural world[1]. The phenomena are built by
the oscillation phenomena, and two things can synchronize
when the two things have almost same natural frequency
and oscillators can aﬀect each other even if distance between the two things is far. Therefore, the synchronization phenomena are critically needed to the communication
systems. Furthermore, the synchronization phenomena arevery close phenomenon to living things and one of essential
phenomena for people living. For instance, in south-east
Asia, it is observed that blink of a large number of ﬁreﬂies
on a tree synchronize. It is considered important to observeand to analyze such synchronization phenomena in order toknow and use various phenomena in this natural world.
In our previous study, we have been observing and analyzing the synchronization phenomen on the coupled oscillators systems which are ring, ladder and 2D lattice shapes.
We discovered a propagating wave which is continuously
switching phase states between two adjacent oscillatorson the ladder from in-phase synchronization to anti-phasesynchronization or anti-phase synchronization to in-phasesynchronization[2]. We named this phenomenon a phaseinversion wave. The phase-inversion waves can be observed on the van der Pol oscillators coupled as the ladder, the ring, the 2D lattice, and so on, when nonlinearity
is weak. We discovered another wave which disappears
or becomes phase-inversion waves in propagating[3]. The
phenomenon is named phase-waves. We have been investigating and analyzing the phase-inversion waves andthe phase-waves by using instantaneous electric power of
each oscillator, instantaneous frequencies, and phase diﬀerences between adjacent oscillators. The phase-waves are
hard to distinguished for phase-inversion waves in shorttime, and the phase-waves are hard to observe when thephase-waves is propagating small phase diﬀerence. Differences in each phenomenon have been conﬁrmed using
the instantaneous electric powers. However, minutia diﬀerence between phase-waves and phase-inversion waves was
not enough analyzed. The phase-waves which is propagating small phase diﬀerences are hard to observe, but it isguessed that the small phase-waves aﬀect other wave phe-

L0
i1

L0

L0

i2
C

ik
C

v1

L1

f(v2)

C

vk

L1

f(vk)

Oscillator 2

Oscillator 1

iN
C

v2

L1

f(v1)

L0

L0

vN

L1

f(vN)

Oscillator k

Oscillator N

Figure 1: Circuit model.
nomena. Therefore, the small phase-waves have to be investigated and analyzed
In this study, we deﬁne as an instantaneous expanding
rate (IER) a change rate of a value of voltage on Poincare
section of which a current is zero in each oscillator, anda quasi Lyapunov exponent (QLE) is deﬁned by using behavior of the voltage on the Poincare section. We suggest toinvestigate and to analyze by using the IER and the QLE.
The phase-inversion waves and phase-waves are observedby using the IER and the QLE on the ladder of van der
Pol oscillators. The phase-waves propagating small phase
diﬀerence are observed.
2. Circuit Model
Our circuit model, which N van der Pol oscillators arecoupled by inductors as a ladder, are shown in Fig.1. Eachvan der Pol oscillator is called OSCk :(1 ≤ k ≤ N). An
inductor and a capacitor of each van der Pol oscillator are
shown as L1 and C, respectively. A voltage of OSCk is
expressed as vk and an electric current in L1 is expressed as
ik . A non-linear negative resistor is expressed as Eq. (1).
f (vk ) = −g1 vk + g3 v3k (g1 , g3 > 0)(1)

<Normalized Circuit Equation>
Circuit equations are normalized by Eq.(2). The normalized equations are shown in Eqs. (3)–(5).
s
r
p
Cg1
g1
xk , vk =yk , t = L1Cτ,iL1 =
3L1 g3
3g3
(2)
r
g21
L1
L1 d
α = , ε = g1,
= “ · ”, δ =
.
L0
C dτ
3g3
<1 st Oscillator (k = 1)>x˙k = yk ,1
y˙k = −xk + α(xk+1 − xk ) + ε(yk − y3k ).
3(3)<Middle Oscillators (1 < k < N)>
x˙k = yk ,- 46 -

1y˙k = −xk + α(xk+1 − 2xk + xk−1 ) + ε(yk − y3k ).
3(4)

<Nth Oscillator (k = N)>x˙k = yk ,1(5)
y˙k = −xk + α(xk−1 − xk ) + ε(yk − y3k ),3where α is strength of coupling, and ε expresses strengthof non-linearity. We analyze these circuit equations by the
fourth Runge-Kutta method. We ﬁx stepsize of RungeKutta method is 0.001 in this study.
3. Instantaneous Expanding Rate andQuasi Lyapunov exponent
We obtain a Poincare section, and an instantaneous expanding rate (IER) and a quasi Lyapunov exponent (QLE)
are deﬁned by using numerical simulations.
3.1. Poincare section
We calculate yk on Poincare section of when xk is zero.
We assume l-th calculation results are xk (l) and yk (l), andthe time is assumed as τ(l). (l + 1)-th calculation results
express xk (l + 1) and yk (l + 1), and the time expresses τ(l +1). We assume the xk (l) is larger than 0, and the xk (l +1) is smaller than 0. A value of yk on Poincare section is
calculated by Eq. (6). The value is shown as ypk (l).
ypk (l) = yk (l) + sign ·xk (l) · |yk (l + 1) − yk (l)|.
xk (l) − xk (l + 1)

(6)

{If yk (l+1) ≥ yk (l), sign = 1. If yk (l+1) < yk (l), sign = −1.}
The time τpk (l) of ypk (l) is assumed by Eq. (7).
τpk (l) = τk (l) +xk (l) · (τk (l + 1) − τk (l)).
xk (l) − xk (l + 1)

(7)

3.2. Instantaneous expanding rate (IER)
The l-th time value on Poincare section is shown as
ypk (l), and the value of (l + 1)-th time is expressed asypk (l + 1). The IER is deﬁned as Eq.(8).
ypk (l + 1) − ypk (l)IER(l) =.
(8)
ypk (l)3.3. Quasi Lyapunov exponent (QLE)
In this paper, below value is called quasi Lyapunov exponent, because QLE is calculated by considering only xk
and yk in OSCk without other oscillators. The QLE is calculated by Eq.(9).
1 X ypk (l + 1) − ypk (l)
ln
.
m l=1
τpk (l + 1) − τpk (l)
m

QLE =

(9)

4. Phase-Inversion Waves and Phase-Waves
The IERs and the QLEs of the phase-inversion waves
and the phase-waves are investigated. The observation conditions are set as follows.
1. N is ﬁxed 200.
2. Strength of the non-linearity ε, and strength of thecoupling α are ﬁxed as 0.3 and 0.05, respectively.
3. Maximum calculation time is 483000τ.
4. We observe two type phase-inversion waves. First
type is that time between ﬁrst and second phaseinversion waves is very short. Second type is that thetime between ﬁrst and second phase-inversion waves
is around 650τ. We call the two type as thin-typephase-inversion waves and wide-type phase-inversion
waves, respectively.

When the synchronization states are in-phase, the QLE isaround -18.15. The QLE is around -17.87 when anti-phasesynchronization is observed. We can guess that QLE is minus inﬁnity in theory, but QLEs are obtained above values
because numerical simulation is used.
4.1. Wide-type phase-inversion waves<Phase state>
Figure 2(a) shows itinerancies of phase states between
adjacent oscillators from 0τ to 16100τ. The Fig. 2(a)shows two hundred horizontally long slim boxes which are
piled up vertically. Horizontal axis of each slim box expresses time. Vertical axis of each slim box expresses a sumof voltages of adjacent oscillators. For instance, the top
slim box shows an itinerancy of phase state between OSC0and OSC1 along time. The next slim box shows an itinerancy of phase state between OSC1 and OSC2 along time.
The bottom slim box shows an itinerancy of phase state
between OSC198 and OSC199 along time. A vibration islarge in the slim box, when a phase states between adjacentoscillators is in-phase synchronization. When the phasestate between adjacent oscillators is the anti-phase synchronization, the vibration decreases to zero. Therefore, black
regions show near in-phase synchronization and white regions show near anti-phase synchronization. We can understand that the second phase-inversion wave, which changes
from anti-phase synchronization to in-phase synchronization, propagate after the ﬁrst phase-inversion wave, whichchanges from in-phase synchronization to anti-phase synchronization, propagates. The time between ﬁrst and second phase-inversion waves is almost 650τ.
<Instantaneous expanding rate>
Figures 2(b)-(e) show the result of IERs. Vertical axis
expresses IER. Horizontal axis expresses time. Itinerancies of IERs are piled up and shown in the Figs. 2(b) and(c). Top line expresses itinerancy of IER0 of OSC0 bottom
graph expresses IER199 of OSC199 . The Fig. 2(b) showsitinerancies of IERs during 483000τ. The Fig. 2(c) is extended ﬁgure of the Fig. 2(b) from 0τ to 16100τ. Figures 2(d) and (e) shows extended ﬁgures of a yellow rectangle box of the Fig. 2(c). The Fig. 2(d) shows behaviorof IER50 of when a phase state is changed from in-phasesynchronization to anti-phase synchronization by the ﬁrst
phase-inversion wave. The Fig. 2(e) shows behavior of
IER50 of when a phase state is changed from anti-phasesynchronization to in-phase synchronization by the second
phase-inversion wave. When a phase-inversion wave propagates, IER50 slightly increase, decrease, increase and become zero again. In other words, two peaks for positive
values and one peak for negative value can be observed,
when a phase state is switched in-phase and anti-phase.
<Quasi Lyapunov exponent>
Figure 2(f) shows QLE50 . Vertical axis expresses QLE.
Horizontal axis is time. QLE asymptotically becomes toaround -13.
4.2. Thin-type phase-inversion waves<Phase state>
Figure 3(a) shows itinerancies of phase states between
adjacent oscillators from 0τ to 16100τ. We can observe thethin-type phase-inversion waves.

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5. Conclusion
We investigated behaviors of IERs and QLEs of whenphase-inversion waves or phase-waves were propagating.
One of diﬀerences between phase-inversion waves andphase-waves was clariﬁed by using IERs. It was clariﬁedthat details of motion of phase-waves can be observed. The
QLEs asymptotically became to about -13. In these results,
we can say that it is eﬀective approach to analyze phaseinversion waves and the phase-waves by using IERs and
QLEs.
References[1] L. Lin, F. Li and M. Ma, “Synchronization of bionanomachines nased on molecular diﬀusion,” IEEE
Sensors Journal, vol. 16, issue 16, pp. 7267-7277,
2016.
[2] M. Yamauchi, M. Wada, Y. Nishio and A. 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.
[3] M, Yamauchi, Y. Nishio and A. Ushida, “Phase-waves
in a ladder of oscillators,” IEICE Trans. Fundamentals,
vol. E86-A, no. 4, pp. 891-899, Apr. 2003.

(a)
y198
+y199

(c)

(d)

16100τ

0τ

483000τ0τ

16100τ

10787τ

11270τ

11431τ
-12-13

11914τ

Instantaneous Expanding Rate0τ

Instantaneous Expanding Rate(b)

Instantaneous Expanding Rate Instantaneous Expanding Rate

4.3. Phase-waves<Phase state>
Figure 4(a) shows itinerancies of phase states from 0τ to
16100τ. Details of phase-waves are hard to observe on thisﬁgure.
<Instantaneous expanding rate>
Figures 4(b), and (c) show the IERs. The Fig. 4(b)shows all itinerancies of IERs. The Fig. 4(c) shows a extended ﬁgure of yellow box of the Fig. 4(b). A range ofvertical axis of the Fig. 4(c) is almost 0.01 times of theFigs. 2(d) and 3(d). We can understand that ﬂuctuationis very small, and form of behavior in Fig. 4(c) diﬀers
forms of behaviors in the Figs. 2(d) and 3(d). There are
two peaks of positive values and two peaks of negative values. Furthermore, the phase-waves in the Fig. 2(a) is hardto observe, but the phase-waves in the Fig. 2(b) is shown
clear. We can understand that there are many very small
phase-waves.
<Quasi Lyapunov exponent>
Figure 4(d) shows QLE50 . We can observe that QLE
asymptotically becomes to around -13.

y0+y1

0.03

0

-0.03

(e)

0.03

0

-0.03

(f)

Quasi Lyapunov Exponent<Instantaneous expanding rate>
Figures 3(b)-(d) show the IERs. The Fig. 3(b) showsitinerancies of IERs from 0τ to 483000τ. The Fig. 3(c)shows a extended ﬁgure of yellow box of the Fig. 3(b).
The Fig. 3(d) shows a extended ﬁgure of yellow box of
the Fig. 3(c). The Fig. 3(d) shows itinerancy of IER50
from 11431τ to 11914τ. We can observed three peaks ofpositive values and two peaks of negative values in the Fig.
3(d). We can understand that second peak of positive value
by ﬁrst phase-inversion wave is combined ﬁrst peak of positive value by second phase-inversion wave, because time
between the ﬁrst and the second phase-inversion waves isvery short.
<Quasi Lyapunov exponent>
Figure 3(e) shows QLE50 . The Fig. 3(e) is similar to theFig. 2(f). QLE asymptotically becomes to around -13.

-18
-19
0τ

Time

483000τ

Figure 2: Wide-type phase-inversion waves. (a) phasestates (0τ to 16100τ), (b) whole of IERs, (c) behaviorof IER50 (0τ to 16100τ), (d) behavior of IER50 (10787τ
to 11270τ), (e) behavior of IER50 (11431τ to 11914τ), (f)QLE50 .

- 48 -

y0+y1

y0+y1

(a)

(a)

y198
+y199

y198
+y199

(b)

16100τ

0τ

0.03

0

(c)

Quasi Lyapunov Exponent

0τ

483000τ

-18
-19

Time

0

-0.000215295τ
-12-13

11914τ

(d)0τ

16100τ

0.0002-0.03
11431τ
-12
-13

(e)

0τ

Instantaneous Expanding Rate

Instantaneous Expanding Rate(d)483000τ

Instantaneous Expanding Rate(c)

Instantaneous Expanding Rate0τ

483000τ

Figure 3: Thin-type phase-inversion waves. (a) phasestates (0τ to 16100τ), (b) whole of IERs, (c) behaviorof IER50 (0τ to 16100τ), (d) behavior of IER50 (11431τ to11914τ), (e) QLE50 .

Quasi Lyapunov Exponent(b)

16100τ

Instantaneous Expanding Rate

0τ

-18
-19
0τ

233450τTime483000τ

Figure 4: Phase-waves. (a) phase states (0τ to 16100τ), (b)whole of IERs, (c) behavior of IER50 (15295τ to 233450τ),(d) QLE50 .

- 49 -