Detailed Analysis on Phase-Waves Generated in a Ladder ofCoupled van der Pol Oscillators

Masayuki YAMAUCHI, Yoshifumi NISHIO and Akio USHIDA

Dept. of Electrical and Electronic Engineering,
Tokushima University, Tokushima 770-8506,Japanfmasa,nishio,ushidag @ee.tokushima-u.ac.jp

Abstract| In this study, wave propagation phenomena of phase di erences observed in van der Pol
oscillators coupled by inductors as a ladder are investigated. For the case of 17 oscillators, interesting phenomena are found. It is called \phase-wave." Thewave changing behavior from in-phase to anti-phaseor anti-phase to in-phase is called \phase-inversionwave." We observe that the behavior of the phasewaves generated by giving phase di erence of plus
value is di erent from those generated by giving phase
di erence of minus value. We analyze the di erence indetail.
I. INTRODUCTION

Large number of coupled limit-cycle oscillators are useful as models for a wide variety of systems in naturalelds, for example, diverse physiological organs including gastrointestinal tracts and axial ber of nervoussystems, convecting uids, arrays of Josephson junctions and so on. Hence, it is very important to analyze
synchronization and the related phenomena observedin coupled oscillators in order to clarify mechanismsof generations or in order to control the generatingconditions of various phenomena in such natural systems. In the eld of the electrical engineering, a lot of
studies on synchronization phenomena of coupled vander Pol oscillators have been carried out up to now.
Recently, we have discovered very interesting wave
propagation phenomena of phase states between twoadjacent oscillators in an array of van der Pol oscillators coupled by inductors [1]. In the study, wenamed the continuously existing wave of changing
phase states between two adjacent oscillators from inphase to anti-phase or from anti-phase to in-phase as\phase-inversion-wave" and explained their generation
mechanisms by using the relations between the oscillation frequencies and the phase di erences of two adjacent oscillators. We also investigated what kind ofphenomena were observed when two phase-inversionwaves collide in the middle of the array [2]. The observed phenomena was classi ed into three types; completely extinction, completely re ection and interme-diate complex phenomenon. Further, the phenomena
related with the re ection and the penetration of thephase-inversion-waves in the system including discontinuity were investigated [3]. Behavior of the phaseinversion-waves around a discontinuity were classi ed
into eight types by computer simulations.
In this study, we report the detail on \phase-wave,"which is propagation of the phase di erence less than
180 degrees between two adjacent oscillators [4]. The
phase-waves exist only in the transient states unlike
the phase-inversion-waves. In the computer calculations, we produce phase-waves as follows. Almost
same initial conditions are given for all oscillators to
produce complete in-phase synchronization, which is
one stable steady states in the system. After the system settles in the complete in-phase synchronization,
the voltage and the current of the arbitrary phase difference is input in the rst oscillator. We observe thatthe behavior of the phase-waves generates by giving
phase di erence of plus value is di erent from thatgenerates by giving phase di erence of minus value.
We analyze the di erence in detail.
II. CIRCUIT MODEL

Circuit model used in this study is shown in Fig. 1. N
van der Pol oscillators are coupled by coupling inductors L0. We carried out computer calculations for thecases of N = 17. Although we introduce the results
only for 17 oscillators, the similar phenomena are observed from both of even and odd. In the computer
calculations, we assume the v 0 i characteristics of thenonlinear negative resistors in the each circuit as thefollowing function.
i (v ) = 0g1 v + g3v 3 (g1 ; g3 > 0)(1)
The circuit equations governing the circuit in Fig. 1
are written as[First Oscillator]
x_ 1 = y1(2)1
y_1 = 0x1 + (x2 0 x1 ) + " y1 0 y133rkkk

τ k (n)Vn-1

OSC kn+1n-2n+3n+4n+2nTime
τ k+1 (n)Vn-1

OSC k+1n-2n+1nn+3n+2n+4Time

Figure 2: De nition of phase di erence.
In upper diagrams of Fig. 3, vertical axis is the sumof voltages of adjacent oscillators, and horizontal axis
Figure 1: Coupled van der Pol oscillators as a ladder. is time. Hence, the diagrams show how phase di erences between adjacent oscillators change as time goes.
[Middle Oscillators]
In lower gures, snapshots of attractor of each oscillax_ = y (3)1 tor and phase states between adjacent oscillators arey_ = 0x + (x +1 0 2x + x 01 ) + " y 0 y 3 shown respectively.
3(k = 2  N01)
B. CLASSIFICATION[Last Oscillator]
phase-waves are generated by phase di erencex_ = y(4) Whenofplusvalue or minus value at an edge of the ladder,
1y_ = 0x + (x 01 0 x ) + " y 0 y 3
observedphenomena are classi ed into four patterns
3by input phase di erence as follows:where<plus value>srg
pCg1
1 y ; (a) around 0  +69[degree]: The phase di erencet = L1C ; i 1 =x ;v =3Lg
3
g
1
3
3
decreases while the phase-waves are propagating.
rL1
L1 d
The phase-waves become extinction at the end of= L ; " = g1 C ; d = \ 1 " :(5)the array.
kkkkNNNkNkNkk

NL kNk0kkNk

It should be noted that corresponds to the couplingof oscillators and " corresponds to the nonlinearityof oscillators. Simulation are calculated by using thefourth-order Runge-Kutta method with Eqs. (2){(4).
III. PHASE-WAVES

A. PHASE-WAVES(b) around +70[degre]]: The phase di erence increases while the phase-waves are propagating.
After the phase-waves re ect at the end of thearray, they become extinction while propagating.
Because the phase-waves do not have enoughphase di erences to become anti-phase at theedge of the array.
(c) around +71  +171[degree]: The phase di erence increases while the phase-waves are propagating. The phase-waves re ect at the edgeof the array and change to the phase-inversionwaves, because the phase-waves have the phase
di erences to become anti-phase at the edge ofthe array.
(d) around +172  +180[degree]: The phase-waves
change to the phase-inversion-waves while thephase-waves are propagating.

In this section, wave propagation phenomenon observed from the circuit with 17 oscillators is investigated. Further, we x = 0:050 and " = 0:30.
Throughout the paper, we de ne the phase di erence between two adjacent oscillators as follows:(n) 2 8 +1 (n) =  ((nn))00 (+1n 0 1)
where  (n) is time when the voltage of OSCk crosses
0[V] at n-th time.
Fig. 3 show typical examples of observed phasewaves which are propagation of the phase-di erence
between adjacent oscillators. These results are obtained for the same parameter values by changing ini- <minus value>
tial conditions as follows:(a) around 0  067[degree]: The phase di erence
decreases while the phase-waves are propagating.
1. Setting as initial conditions of all oscillators theThe phase-waves become extinction at the end ofsame.
the array.
2. Putting as the arbitrary phase di erence of the(b) around 068[degree]: The phase di erence involtage and the current of one oscillator.
creases while the phase-waves are propagating.
kkk;kkkk

The phase-waves re ect at the end of the array and change to the phase-inversion-waves, because the phase-waves have the phase di erences
to become anti-phase at the edge of the array.
(c) around 069  0180[degree]: The phase di erence increases while the phase-waves are propagating. The phase-waves become the phaseinversion-waves before which arrive the edge ofthe array.
(d) around 059  085[degree]: Two pair of waves
are generated.
We observed that the behavior of the phase-waves
generated by giving phase di erence of plus value isdi erent from that generated by giving phase di erence of minus value. We analyze the di erence indetail. Further, when the phase-waves are generated (i) input phase di erence +60[deg](domain plus-(a))
by giving phase di erence of minus value, we foundthe phenomena that phase-waves of two sets generate.
Fig. 4 shows domain of the observed phenomena.
IV. CONCLUSION

In this study, the phase-waves were investigated bynumerical calculation. We found that the behavior of
the phase-waves generated by giving phase di erenceof plus value is di erent from that generated by giving
phase di erence of minus value. We analyze the difference in detail. Further, when the phase-waves aregenerated by giving phase di erence of minus value,
we found the phenomena that phase-waves of two setsgenerate.
References[1] M.Yamauchi, M.Wada, Y.Nishio and A.Ushida,\Wave Propagation Phenomena of Phase States (ii) input phase di erence 050[deg](domain minus-(a))
in Oscillators Coupled by Inductors as a Ladder," IEICE Trans. Fundamentals, vol.E82-A, no.11,
pp.2592-2598, Nov. 1999.
[2] M.Yamauchi, M.Wada, Y.Nishio and A.Ushida,\Collisions between Two Phase-Inversion-Waves
in an Array of Oscillators,"
Proc. ISCAS'00, vol.1, pp.679-682, May 2000.
[3] M.Yamauchi, Y.Nishio and A.Ushida, \Re ection and Transmission of Phase-Inversion-Waves
in Oscillators Coupled by Two Kinds Inductors,"
Proc. ISCAS'01, vol.3, pp.767-770, May 2001.
[4] M.Yamauchi, M.Wada, Y.Nishio and A.Ushida,\Analysis of Phase Di erence Propagation in Oscillators Coupled by Inductors as a Ladder," Proc.
NOLTA'99, vol.2, pp.783-786, Nov. 1999.
(iii) input phase di erence +70[deg](domain plus-(b))
Figure 3: Phase-waves. = 0:05; " = 0:3.

(iv) input phase di erence 068[deg](domain minus-(b) (vii) input phase di erence +175[deg](domain plusand minus-(d))(d))(v) input phase di erence +135[deg](domain plus-(c)) (viii) input phase di erence 070[deg](domain minus(d))
Figure 3: (continued).

The phase-waves are generated by the phase
di erence of plus value.

The phase-waves are generated by the phase
di erence of minus value.
(vi) input phase di erence 0135[deg](domain minusFigure 4: Domain of observed phenomena that are(c))
Figure 3: (continued).
gerated by the phase di erence of plus or minus value.