OF PHASECONTROL

STATES IN RING

Yoshinobu

SETOU,Yoshifumi

OSCILLATORSNISHIOand

INDUCTORSSHARING

Akio USHIDA

Department of Electrical and Electronic Engineering, Faculty ofEngineering, Tokushima University,Tokushima, 770, JAPANTEL: +81-88656-747Q FAX: +81-886-56-7471
Email: setou@ee.tokushima-u.ac.jp

ABSTRACTIn this study we show control of phase states in rinrz Oscillators sharing inductors by infus~g the voltage of the firstoscillator into the other one. We conlirro the suppression ofthe phzae states of the original system and the generationof new phase states which we have not seen before infusing
the voltage.

1.

expected to be one of the future parallel information processing architectures.
2,CIRCUIT

MODELFig. 1 shows our method. By using the switch, it is possible to infuse the voltage of the iirst oscillator into the i-thoscillator. The synchronization phenomena of the original
system without infusing haa reported in [13].

INTRODUCTION~.-Since coupled oscillator systems can describe various ph~nomena in natural fields, there have been many inveAigations on such systems and various interesting phenomena
have been reported by several researchers. For example,
Mori et al and Kimnra et al have investigated synchronization phenomena observed in two oscillators coupled by acapacitor [1] and a resistor [2] respectively. Endo et af andNiahio et al have investigated synchronization phenomena
observed in many oscillators coupled by inductors [3]-[5] andresistors [6]-[8]. We have confirmed the oscillation deathin many oscillators coupled by resistors [9][10]. However,there are not many investigations on oscillators sharing elements. Scott has investigated synchronization phenomena in l-dimensional (ID) and 2-dimensional (2D) arraysof almost-linear and almost lossless networks [11][12]. He
has only treated these system as distributed circuits. We
investigated synchronization phenomena in oscillators sharing inductors [13].
On the other hand recently many researchers pay theirattentions to the possibility of.application to parallel information
processing architecture using coupled oscillators or
coupled chaotic circuits. As the fist step for this purpose,
we need to control phase states in coupled oscillators byany methods. Namely, we have to restrict generating phasestates, make some oscillators stop, or produce new previous
unseen phase states.
In this study we investigate control of phase states in~g osc~ators shfing inductors with *singvoltage Of
the fist oscillator into the other one. In this system thevoltage of the tirst oscillator is infused into the other one
through the bufkr so M to control phase states. We can
expect the generation of interesting phase states by infusing voltage because all oscillators which are synchronized
at in-phase are unstable in the original system. We confirm
the suppression of the phase states in the original system
without voltage infusion and the generation of new previously unseen phase states after infusing the voltage. Thesephenomena are confirmed by both numerical calculations
and circuit experiments. The present study contributes tothe development of coupled oscillators networks which are/.-.“““”LI\\.....~\I/

Lnul-lklea*M

Fig. 1 Circuit model.
We assume that v-i characteristics of the nonlinear resistor is represented by third order polynomial equationi,(v) = -g*v+gsv3(1)( gl, g3 > 0).

Changing the variables

t=~T,Vk=.@zk,‘k=@Yk(2,and definingL(3)[ Fthen the normalized equations are represented as follows
E=g~~k =‘Yk+ Yk-1+S13(Zk –(4)~~k)~k = Zk - zk~l(5)(k=l,2,3,...,Nexcept

k=i).

The equations of i-th infused oscillator areii =‘Yi+vi-l

Yi-1 = Zi-1 -21

Copyright 1997 IEEEji =21— Zi+l+~(21-13(6)~Zl)

(7)
(8)

813

where e is the largeness of nonlinearity. For numerical calculations, to consider the difference between the natural
frequencies of the real oscillators, the symmetry of the normalized equation (3) is broken by using additional parameters Ak = 0.001k as follows
ik ={‘!/k + Yk-1 +~(k=l,2,3,...,((1+Zk - ~dA:)

(9))}

N).

For the following numerical calculations, the largeness ofnonlinearity is c = 0.05. For circuit experiments, we use
the parameters of C = 33[nF] and L = 10[mH].
3.

NUMERICALCIRCUIT

W,wl,

4.1

CALCULATIONSEXPERIMENTS

AND

3.1.

In the case of N = 4
In our previous work we have conii.rmed the generation of
two types of phsse states [13]. One of them is shown in Fig.
2 where phase diEerence between two adjacent oscillators
is x. The other is shown in Fig. 3. For this case phasediihrence between diagonal oscillators is x, but two pairs
are not synchronized. The generation of these two phasestates are decided by the initial condition.
When we infuse the voltage of the first oscillator into thethird one, the phase state of Fig. 3 is suppressed and only
the phase state of Fig. 2 generates. Namely it shows that
we can restrict the phase states by infusing the voltage.

(b)
Fig.
3 Phase state observed only for the case without
infusing ( N = 4 ).
(a) Numerical calculation.
(b) Experimental result( Vertical 1 V/div. Horizontal 20 ps/div. ).

IL(a) +T(b)

Fig. 4 New phase state observed for the case with infusing
into the second oscillator ( N = 4 ).
(a) Numerical calculation.
(b) Experimental result( Vertical 1 V/div. Horizontal 20 ps/div. ).
3.3.
(a) +T(b)

Fig. 2 Phase state observed for both cases without infusing and with infusing into the third oscillator ( N = 4 ).
(a) Numerical calculation.
(b) Experimental result( Vertical 1 V/div. Horizontal 20 ps/div. ).
On the other hand we can get the new phase state shownin Fig. 4 by infusing the voltage of the fist oscillator into
the second one. This phase state have never seen before we
had infused the voltage. In the phase state the first andthe second oscillators are synchronized at in-phase and thethird and the fourth oscillators are synchronized at opposite
phase to the first one.
3.2.

In the case of N = 8

Similarly to the N = 4 we have confh-med the generation ofphase states extending Figs. 2 and 3 [13]. When we infuse
the voltage of the first oscillator into the third one, we canonly observe the phase state in Fig. 5 by suppression tothe other phase states. On the other hand we confirm thegeneration of the new phase state by infusing the voltage ofthe ii.rst oscillator into the second one shown in Fig 6.
We can not observe the new phase state by infusing thevoltage of the fist oscillator into the third, the fourth or thefifth one. In that case the system can not be synchronized.

Inthecaaeof N=50r

N=7

We have confirmed the generation of two types of 5 phasestates [13]. One is the phase shift of 2x/5 between adjacentoscillator as in Fig.7 and the other is the phase shift of 4T/5.
These phase states disappear by infusing the voltage of thefirst oscillator into the other one.
We confirm the generation of the new phase pattern byinfusing the voltage of the fist oscillator into the second one
shown in Fig. 8. For the phase state in Fig. 8 the fist, thesecond and the third oscillators are synchronized at in-phzse
and the fourth and the fifth oscillators are synchronized atopposite phase to the fist one. The interest thing is that thephase state in Fig. 8 cannot be observed when we infuse thevoltage of the first oscillator into the third one, neverthelessthe first and the third oscillators are synchronized at inphase.
In the case of N = 7, three types of 7 phase states exist
where the phsse shift is 2u/7, 4x/7 or 6T/7 [13]. Similarlyto the case of N = 5, we cordh-m the suppression to thesephase states and the generation of the new phase states
shown in Fig. 9 by infusing the voltage of the first oscillator
into the fourth oscillator.
From the above two examples when we infuse the voltageof the first oscillator into the other one in the case that N
is odd, we can predict the suppression to the phase states
and the generation of the phase states similar to Fig. 9Copyright 1997 IEEE814

af’vwd‘J&+d“$Ww41+@’ALI1I‘&R5‘J7alJ&%41-~Fig. 5 Phsae state observed for both cases without infusing and with infusing into the third oscillator ( N = 8 ).
(a) Numerical calculation.
(b) Experimental result( Vertical 1 V/div. Horizontal 20 ps/div. ).
4.

OSCILLATION

DEATH

We discovered more interest synchronization phenomenon
for the case of N = 6 by infusing the voltage, namely oscillations of some oscillators stop by infusing the voltage.
The first one shown in Fig. 10(a) is observed by infusing the voltage of the first oscillator into the second one.
The tlrst, the second, the fourth and the fifth oscillator areoscillating. On the other hand the third and the sixth oscillators stop to oscillate. The second shown in Fig. 10(b) isobserved by infusing the voltage of the first oscillator into
the third oscillator. The second, the fourth and the sixth
oscillators are oscillating. On the other hand the fist, thethird and the sixth oscillators stop to oscillate.
However this phenomenon is observed only from the circuit experiments. We consider this is because the nonlinear
resistor is approximated by third order polynomial equationin numerical calculations.
5.

CONCLUSIONSIn this study we investigated control of phase states in ringoscillators sharing inductors with infusing voltage of theIlrst oscillator into the other one. In this system the voltageof the first oscillator is infused into the other one throughthe btier so as to control phase states. We coniirrned thesuppressing the phase states of the original system and thegeneration of the new phase states which we had not seen
before infusing. These phenomena were confirmed by nuFor the fumerical calculations and circuit experiments.
ture problem are the detailed investigation of the oscillation death in the case of IV = 6 and theoretical proof of theobserved phenomena.
REFERENCES[1] H. Kimura and K. Mano, “Some Properties of Mutually Synchronized Oscillators Coupled by Resistances”,

Fig.

-.
(b)

6 New phase state observed for the case with infusing~to the second oscfitor( N = 8 ).
(a) Numerical calculation.
(b) Experimental result( Vertical 1 V/div. Horizontal 20 ps/div.

).

Trans. IECE, Vol. 48, No. 10, pp. 1647-1656, Oct. 1965
( in Japanese).
[2] T. Suezaki and S. Mori, “Mutual Synchronization ofTwo Oscillators”, ‘l+ans. IECE, vol. 48, no.9, pp. 15511557, Sep. 1965 ( in Japanese ).
[3] T. Endo and S. Mori, “Mode Analysis of a multimodeladder oscillators”, IEEE Trans. Cinxits
Syd., Vol.
CAS-23, No. 2, pp. 100-113, Feb. 1976.
[4] T. Endo and S. Mori, ‘Mode Analysis of a two dimensional low-pass multimode oscillators”, IEEE Trans.
Circuits Syst., Vol. CAS-23, No. 9, pp. 517-530, Sep.
1976.
[5] T. Endo amd S. Men, ‘Mode Analysis of a Ring of aLarge Number of Mutually Coupled van der Po1 Oscillators”, IEEE 2Yans. Circuits Syst., Vol. CAS-25, No.
1, pp. 7-18, 1978.
[6] S. Moro, Y. Nishio and S. Mori, ‘Syncbxonization phenomena in oscillators coupled by one resistor”, IEICE
Trans. Fundamentals,
vol. E78-A no. 2 pp. 24.4253,
Feb. 1995.
[7] S. Moro, Y. Nishio and S. Mori, “Synchronization phenomena in RC oscillators coupled by one resistor”, IEICE Trans. Fundamentals, vol. E78-A no, 10 pp. 14351439, Oct. 1995.
[8] Y. Nishio and S. Mori, ‘Mutually Coupled Oscillatorswith an Extremely Lmge Number of Steady States”,
Proc. of ISCAS’9~ , pp. 819-822, May 1992.
[9] Y. Setou, Y. Nishio and A. Ushida, “Synchronization
phenomena in many oscillators coupled by resistors asa ring”, ~mc. OSAPCCAS ’94, pp. 570-575, Dec. 1994.
[10] Y. Setou Y. Nishio and A. Ushida, “Synchronization
phenomena in resistively coupled oscillators with different frequencies”, IEICE ‘Ikan.s. Fundamentals,vol.
E79-A, No. 10 Oct. 1996.

Copyright 1997 IEEE815

[11] A. C. Scott, “Distributed Multimode Oscillators of One
and Two Spatial Dimensions”, IEEE TronJ. Cixuit
Theory , vol. CT-17, No. 1, pp. 55-60, Feb. 1970.
[12] A. C. Scott, ‘The Distributed Tunnel diode Oscillator“, flkans. IRE Circuit Theory , vol. CT-10, pp. 53-59,
March 1963.
[13] Y. Setou Y. Niehio and A. 17shida,—. “On Synchronization Phenomena in Oscillators Sharing Inductors”,
Proc. of NOLTA ’96, pp. 313-316, Ott 1996.

Y.

‘led‘$%a11,

8

I_LL—d+t(b)Fig.

7(a) +Phase state observed only for the case without~siu(~=s).
(a) Numerical calculation.
(b)., Emerimental result( ~ertical 1 V/div. Horizontal 20 ps/div.

“leed‘2523Q :‘+5E9 ‘‘IeEEI :‘SIE53(a)’7T+t(b)Fig. 9 New phiiae state observed for the case with infusing~to the second osc~tor( N = 7 ).
(a) Numerical calculation.
(b) Experimental result( Vertical 1 V/div. Horizontal 20 ps/div. ).

).

❑+t(b)

Fig. 8 New phase state observed for case with infusing
into the second oscillator ( N “= 5 ).
(a) Numerical calculation.
(b) Experimental result( Vertical 1 V/div. Horizontal 20 ps/div. ).

(a)

(b)

Fig. 10 The oscillation death ( N = 6 ).
(a) Infusing the voltage of the first oscillator into the secondone.
(b) Infusing the voltage of the first oscillator into the thirdone.
Vertical 1 V/div. Horizontal 20 ps/div.

Copyright 1997 IEEE816