II

MUTUALLY COUPLED OSCILLATORS WITH
AN EXTREMELY LARGE NUMBER OF STEADY STATES
Yoshifumi NISHIO and Shinsaku MORI
Dept. of Electrical Engineering, Keio University

3- 14-1 Hiyoshi, Kohoku-ku, Yokohama, 223 Japa.n
Tel. +81-45-563-1141 Ext. 3319
Fig. 2 and its v - i characteristics is approximated by thefollowing function.

ABSTRACT : There have been many investigationsof the mutual synchronization of oscillators. In this article, N oscillators with the same natural frequency mutually coupled by one resistor are analyzed. In this systemvarious synchronization phenomena can be observed. Especially, we pay our attention to N-phase osillation. It is
confirmed that our system can stably take ( N - l)! phasestat.es by both of computer calculations and circuit experiment.s.

+ir(vk) = -glVk(91,93> 0)(1)

The circuit equation is described as follows,

c-ddtv k = - i k - i r ( V k )
dik

L-N

=dt

U k - R E i ,3=1

INTRODUCTION
There have been many investigations of the mutual synchronization of oscillators ([I]-[4] and therein). Kimura
et al. have confirmed that two oscillators coupled by resistors are synchronized at opposite phase[l]. Endo andme have analyzed a large number of coupled van der Poloscillators[2]-[4]. In such systems various interesting synchronization phenomena are observed.
In this study, we investigate N oscillators with the same
natural oscillating frequency mutually coupled by one resistor. When the nonlinearity of each oscillator is weak
and N is large, each oscillator cannot synchronize. However, in the case of strong nonlinearity, various modesof synchronization can be stably observed. Especially,
we pay our attention to N-phase oscillation. We takethe waveform observed in one oscillator as reference signal. Then, other oscillators can take any phase differencesamong $k = 2k7r/N (k = 1 , 2 , . .. N - 1). Therefore, oursystem can stably take ( N - l)! phase states. This meansthat when N = 13, our system can take 479001600 steadystates. Therefore, our system may be utilized as an extremely large memory.

CIRCUIT MODEL
The circuit model is shown in Fig. 1. The nonlinear
resistor with strong nonlinearity is realized as shown in

By changing the variables,t = mr,(3)(2) is normalized as(4)

wheref(Zk) =-p (zk -$).

(5)

N-PHASE OSCILLATION
At first, we consider the case of N = 5. Figs. 3 and 4
show an example of numerical results obtained by using
the Runge-Kutta method and the corresponding experimental results, respectively. For computer calculations, in

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nonlinear resistor

Fig. 2 Realization of the nonlinear
resistor with strong nonlinearity.

Fig. 1 Circuit model.
order to consider the difference between the natural frequencies of real oscillators, (4)is rewritten as follows.

ANOTHER SYNCHRONIZATIONS
For the case that N is not a prime number, N-phaseoscillation s e e m to be difficult to observe and anothertypes of syncronization are observed. For example, Fig.
7 shows an example of experimental results for the caseof N = 4. In this case, x 1 and x4 almost synchronize
at opposite phase and x 2 and x 3 almost synchronize at
opposite phase. However, the phase between z1 and 2 3
is independent. This result is similar to that in [3]. The
combination of the synchronized oscillators seem to be decided by the differences between the natural frequencies ofreal oscillators.
Similarly, in the case of N = 6, three pairs of two almost
synchronized oscillators are observed. For example, X I
and x 2 , x 3 and x 4 , and 2 5 and x6 almost synchronize,respectively. However, the phase between X I and x 3 , thephase between x, and x 5 and the phase between 2 3 andx 5 are independent. Moreover, in this case we can observefrom the circuit experiments that two independent 3-phaseoscillations are excited.
We consider that our system tends to minimize the current through the coupling resistor. Therefore, in the casethat N is a prime number, N-phase oscillation stably excited. While in the case that N is not a prime number, various combinations to minimize the current ca.n exist and( k l , 2, . ”, N )where A W k corresponds to the difference between the natural oscillating frequency of the reference oscillator andthose of another oscillators.
We can observe that $phase oscillation is stably excited.
In Figs. 3 and 4 each oscillator’s waveform are ordered as( . ~ 1 , ~ 4 , ~ 2 , ~ 5 ,Since~ 3 ) . z1 is reference signal, the numberof the states of za IV x 5 is equal to (5 - l)! = 24. We confirmed that ( 5 - l)! = 24 steady states are stably excitedby both of computer calculations and circuit experiments.
Next, we consider the case of N = 7. Figs. 5 and 6 show
a example of numerical results and the corresponding experimental results, respectively. For the case of strong
nonlinearity, namely for large p, 7-phase oscillation is stably observed. In Figs.5 and 6, each oscillator’s waveform
are ordered as ( 1 1 , x 6 , 2 5 , x 2 , x 3 , x 7 , 2 4 ) . We do not checkthat all of (7 - l)!= 720 steady states are stably excited,
but a large number of steady states are stably observed
from both of computer calculations and circuit experiments.

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,.
.- .-.

IT

Fig. 5 Numerical result for the case of N = 7(a= 2.0, /3 = 5.0, Awl = 0, A w =~ 0.001,
A w=~ 0.002, A w =~ 0.003, A w =~ 0.004,
AWS= 0.005, Aw, = 0.006).

Fig. 3 Numerical result for the case of N = 5.
(a= 2.0, p = 2.0, Awl = 0, A W=~ 0.001,
Aw=~ 0.002, A w =~ 0.003, A w =~ 0.004).

+ t
Fig. 4 Experimantal result for the case of N = 5.
( L = lOmII, C = 0.068pF, R = 2-ilR, horizontal
scale: O.lmsec/div., vertical scale lV/div.)

Fig. 6 Experimantal result for the case of N = 7.
( L = 10mH, C = 0.068pF, R = 241R, horizontal
scale: O.lmsec/div., vertical scale lV/div.)

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CONCLUSIONSIn this study, we have investigated N oscillators withthe same natural oscillating frequency mutually coupledby one resistor. In the case of strong nonlinearity, we canobserve that N-phase oscillations are stably excited fromboth of computer calculations and circuit experiments.
Because this system can take (N - l)!phase states, it may
be utilized as an extremely large memory. Our future research are follows. Experiments for large N. Investigationof another types of synchronization. Theoretical analysis.
Investigation for the case of another types of oscillators.
REFERENCES[I] H. Kimura and K. Mano, ”Some Properties of Mutually

Synchronized Oscillators Coupled by Resistances,” Trans.
IECE, Vol. 48, No. 10, pp. 1647-1656, Oct. 1965 (inJapanese).

Fig. 7 Experimantal result for the case of N = 4.
(I,= IOmII, C = 0.068pF, R = 2410, horizontal
scale: O.lmsec/div., vertical scale lV/div.)[2] T. Endo and S. Mori, ”Mode Analysis of a Multimode Lad-der Oscillator,” IEEE Trans. Circuits Syst., Vol. CAS-23,No. 2, pp. 100-113, Feb. 1976.

various syncronization can be observed. Detailed analysis of such complicated synchronization phenomena is ourfuture research.
If N is a prime number and the nonlinear resistor withstrong nonlinearity is used, N-phase oscillations seems to
be stably excited. Therefore, we believe that we can makea system with an extremely large number of steady statesby using only 13 oscillators.

[3] T. Endo and S . Mori, ”Mode Analysis of a Two-Dimensional Low-Pass Multimode Oscillator,” IEEE
Trans. Circuits Syst., Vol. CAS-23, No. 9, pp. 517-530,
Sep. 1976.
[4] T. Endo and S. Mori, ”Mode Analysis of a Ring of a Large

Number of Mutually Coupled van der Pol Oscillators,”
IEEE Trans. Circuits Syst., Vol. CAS-25, No. 1 , pp.
7-18, Jan. 1978.

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