2004 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'04)
Hawaii, USA, Mar. 5-7, 2004.

Phase-Inversion Waves in Oscillator Ladders Coupled by Inductors
Yusaku ACHI† , Miki AKAZAWA† , Masayuki YAMAUCHI†† and Yoshifumi NISHIO††Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: yusaku@ee.tokushima-u.ac.jp††Sophia University

1. Introduction

N van der Pol oscillators are coupled by coupling inductors L0 . We carried out computer calculations for theIn world today, many kinds of circuits for various cases of N = 8. In the computer calculations, we asapplications have been designed and developed. Espe- sume the v − i characteristics of the nonlinear negativecially, digital circuits use signals generated by oscillators. resistors in each circuit as the following function.
Therefore, oscillators always keep an important position,and various oscillators still continued to be designed and3
(g1 , g3 > 0)(1)
ir (vj,k ) = −g1 vj,k + g3 vj,kput to practical use. Recently, a lot of companies developsystems of high speed. In these systems, high frequency
oscillators are needed. For instance, central processing
units and communication instruments are needed oscillators which generate giga heartz. Especially, in communication, not only oscillators generating high frequency
L 22
Ladder 2
L 22
but also high frequency oscillators which hop at highspeed are needed. A lot of researchers emphasize to deLadder M
velop these oscillators. Moreover, some researches that
L 22
coupled oscillators apply to algorithm of walking of theLadder 1
robot are done [1]-[4].
Recently, we discovered continuously existing wave ofchanging phase states between two adjacent oscillatorsfrom in-phase to anti-phase or from anti-phase to in(a)phase in coupled van der Pol oscillators by inductors asa ladder [5]-[7]. This phenomenon is observed in steady
L 21
L 21
L 21
L 21state. We call this phenomenon as “phase-inversionOOOO
Ladder jwave”. And, the mechanisms of “propagation,” “disSSSSCCCCappearance,” “reﬂection in the middle of the array” and“reﬂection at an edge of the array,” which were the basic
oscillator j,1 oscillator j,2 oscillator j,3oscillator j,Ncharacters of the phase-inversion waves were clariﬁed [5].
(b)In this study, some ladders which are composed of vander Pol oscillators coupled by inductors are coupled byi j,ksome inductors. We investigate various phenomena ofO
L1
C v j,kS
the phase-inversion waves by changing initial values andCparameters. The observed phenomena are classiﬁed andthe relation between the phenomena and the parameters
are clariﬁed.
(c)Figure 1: Circuit Model. (a) Coupled ladders system. (b) Coupled oscillators as a ladder. (c) van der
The circuit model used in this study is shown in Fig. 1. Pol oscillator.

2. Circuit Model- 467 -The circuit equations governing the circuit in Fig. 1 1−(a) The phase-inversion waves propagate to thejoint of ladder and reﬂect. The phase-inversion
are written as:
waves do not penetrate to other ladders and con[Edge Oscillators]
tinuously exist.
(2)
ẋj,1 = yj,1
1−(b) The phase-inversion waves propagate to thejoint of ladder. And the phase-inversion waves
1 3
ẏj,1 = −xj,1 + α1 (xj,2 − xj,1 ) + ε(yj,1 − yj,1)reﬂectat the joint of ladder and penetrate to3
other two ladders. The phase-inversion waves
are continuously existing on all ladders.
[Middle Oscillators]
1−(c) The phase-inversion waves propagate to theẋj,k = yj,k(3)joint of ladder and disappear.
ẏj,k = −xj,k + α1 (xj,k+1 − 2xj,k + xj,k−1 )1 3+ ε(yj,k − yj,k)(k = 2, 3, . . . , N − 1)3[Adjacent Oscillators of Jointed Point]ẋj,N = yj,N(4)ẏj,N = −xj,N + α1 (xj,N −1 − xj,N )α21 3(xc − xN ) + ε(yj,N − yj,N+)
M3
where(a)
Cg1
g1xj,k , vj,k =yj,k ,3L1 g33g3

L1 d
L1L1,= “ · ”.
α1 =, α2 =, ε = g1
L21
L22
C dτ

t = L1 Cτ, ij,k =It should be noted that α corresponds to the couplingof the oscillators and ε corresponds to the nonlinearityof the oscillators. Throughout the paper, we ﬁx N = 8,
α1 = 0.050, ε = 0.250 and ∆τ = 0.01 and calculate(2)-(4) by using the fourth-order Runge-Kutta method.

(b)

3. Phase-Inversion Waves
Value of α2 and oscillator which are generated phaseinversion waves are changed. And we observe phaseinversion waves in this circuit. Further, we ﬁx that a pairof phase-inversion waves is generated at edge oscillators.
Complex phenomena are observed in domain of α2which is not written in tables. These phenomena can
not been classiﬁed.

(c)Figure 2: M = 3. Waves generated from OSC1,1 .

3.1. Coupled Three Ladders
Observed phenomena are classiﬁed into 3 patterns by 3.2. Coupled Four Ladderschanging α2 . Each number of generated at edge of ladder
Observed phenomena are classiﬁed into 4 patterns by
was summarized in Tab. 1. Figure 2 shows an example
changing α2 . Each number of generated at edge of ladderof phase-inversion waves on the circuit.

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Table 1: M = 3.
(a) Waves are generated from OSC1,1 .
α2Classiﬁcation Figure∼0.0111−(a)2(a)
0.300∼0.376
1−(b)
2(b)
0.044∼0.049
1−(c)
2(c)
0.408∼(b) Waves are generated from OSC1,1 and OSC2,1 .
α2Classiﬁcation∼ 0.0191-(a)0.035 ∼ 0.038
1-(b)0.220 ∼0.044 ∼ 0.046
1-(c)(c) Waves are generated from all edges of ladders.
α2Classiﬁcation
In all almost the areas
1-(a)

(a)

was summarized in Tab. 2. Figure 3 shows an exampleof phase-inversion waves on the circuit.
2−(a) The phase-inversion waves propagate to thejoint of ladder and reﬂect. The phase-inversion
waves do not penetrate to other ladders and continuously exist.
2−(b) The phase-inversion waves propagate to thejoint of ladder. And the phase-inversion waves
do not reﬂect and penetrate to other two ladders. The phase-inversion waves continuously
exist on two ladders.
2−(c) The phase-inversion waves propagate to thejoint of ladder. And the phase-inversion waves
reﬂect at the joint of ladder and penetrate toother two ladders.

(b)2−(d) The phase-inversion waves become extinction
while the phase-inversion waves are propagating.
4. In-and-Anti-Phase Synchronization
In this section, we observes the propagation of thephase-inversion wave in the another type of synchronization. Another type of synchronization is “in-and-antiphase synchronization” which in-phase and anti-phaseare existing alternately. In this section, three ladders
were coupled, and put it into the state of “in-and-anti(c)phase synchronization”. When a pair of phase-inversion
waves are generated from OSC1,1 and OSC2,1 , the phaseinversion waves do not reﬂect and penetrate to the other Figure 3: M = 4. Waves are generated from OSC1,1 andladder. The range of this phenomena which observed in OSC1,2 .
this circuit is shown in Tab. 3.

- 469 -Table 2: M = 4.
(a) Waves are generated from OSC1,1 .
α2∼0.0100.085∼0.124
0.188∼0.299
0.043∼0.072
0.288∼Classiﬁcation2−(a)2−(c)2−(d)(b) Waves are generated from OSC1,1 and OSC2,1 .
α2∼ 0.032
0.057 ∼ 0.275
0.295 ∼Classiﬁcation
2-(a)
2-(b)
2-(c)

Figure
3(a)
3(b)
3(c)

Figure 4: M = 3. Waves are generated from OSC1,1 andOSC2,1 .

(c) Waves are generated from OSC1,1 , OSC2,1 andOSC3,1 .
α2∼ 0.032
0.116 ∼References[1] T. Suezaki and S. Mori, “Mutual Synchronizationof Two Oscillators,” Trans of IECE, vol. 48, no. 9,
pp. 1551–1557, Sep. 1965.

Classiﬁcation
2-(a)2-(c)(d) Waves are generated from all edges of ladders.
α2In all almost the areas[2] T. Endo and S. Mori, “Mode Analysis of a Multimode Ladder Oscillator,” IEEE Trans. Circuits andSyst., vol. 23, pp. 100-113, Feb. 1976.

Classiﬁcation
2-(a)[3] S. Ito, H. Yuasa and K. Ito, “Oscillator-MechanicalModel of the Pattern Transition on Quadrupedal
Locomotion Based on Energy Expenditure,” Trans.
of SICE, Vol. 32, no. 11, pp. 1535–1543, Nov. 1996
(in Japanese).

Table 3: M = 3.
Waves are generated from OSC1,1 and OSC2,1 .
α2Figure
0.083 ∼ 0.094
Fig.4[4] A. Fujii, A. Ishiguro and P. E. Hotz, “Generationof Bipedal Locomotion Using a CPG Circuit withNeuromodulation,” Trans. of SICE, Vol. 38，no. 11，
pp. 981–989, Nov. 2002 (in Japanese).

5. Conclusions

In this study, some ladders which are composed ofvan der Pol oscillators coupled by inductors are coupled
by some inductors as a cross. We investigated various
phenomena of the phase-inversion waves such as “reﬂection,” “penetration,” and “disappearance” at a jointof ladders, by changing initial values and parameters.
When two pairs of phese-inversion waves were generated
at each edge of two ladder of coupled four ladders, thephase-inversion waves were do not reﬂect and penetrate
to other two ladders. Further, the phase-inversion waves
which existing alternately were observed, when all oscillators are state of in-and-anti-phase synchronization.

[5] M. Yamauchi, M. Wada, Y. Nishio and A. Ushida,“Wave Propagation Phenomena of Phase Statesin Oscillators Coupled by Inductors as a Ladder,”IEICE Trans. Fundamentals, vol.E82-A, no.11,pp.2592-2598, Nov. 1999.
[6] 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.
[7] M. Yamauchi, Y. Nishio and A. Ushida, “Reﬂectionand Transmission of Phase-Inversion-Waves in Oscillators Coupled by Two Kinds Inductors,” Proc.
ISCAS’01, vol.3, pp.767-770, May 2001.

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