1.  Introduction
Since coupled oscillators systems can describe various phenomena in natural field, there have been many inves- tigations on such systems and various interesting phe- nomena have reported by several researchers[1]-[5]. Kimura et al. investigated synchronization phenomena observed in two oscillators coupled by a resistor shown in Fig. 1 [ 1 ]. They confirmed that these oscillators were synchronized at t the opposite phase.  They also com- mented that the synchronization seemed to occur so that the system tends to minimize the energy consumed by the coupling resistor. This study was also extended at the case of three oscillators and they confirmed the gen- eration of three phase synchronization [2]. Moreover, we have investigated many oscillators coupled by resis- tors[4]. In [4], we confirmed that for the case of four oscillators the system is synchronized at opposite phase in pairs, and the phase shift between two pairs is not decided. However, these studies treat only the case that all of oscillators have the same oscillation frequency. Of cause, there may be many systems modeled as coupled oscillators with the same oscillation frequency. How- ever, because there are very few coupled systems with the exactly same oscillation frequency in natural field, various synchronization phenomena which cannot be explained by coupled systems with the same frequency exist. Although some studies treating coupled oscilla- tors systems with different oscillation frequencies have been reported, we consider that many unknown nonlin- ear phenomena remain in such systems. Therefore, it is very important to investigate such systems. In this study, we consider some oscillators with different oscillation frequencies coupled by a resistor. Because it is difficult to treat general case, we concen- trate the simplest case that oscillation frequency of only one oscillator is varied. Namely, N - 1 oscillators have the same oscillation frequency and only the N th os- cillator has different frequency.  At first we consider non-resonance. By carrying out circuit experiments and computer calculations, .we observe that oscillation of the Nth oscillator stops in some range of the frequency ratio and that others are synchronized as if the Nth oscillator does not exist. These phenomena are also analyzed theoretically by using the averaging method. Secondly, we investigate the resonance region where the frequency ratio is nearly equal to 1. For this region we can observe interesting double-mode oscillation, that is, synchronization of envelopes of the double-mode oscil¬ lation and change of oscillation amplitude of the Nth oscillator. Because the averaging method cannot be ap¬ plied to the double-mode oscillation, these phenomena are investigated by both of circuit experiments and com- puter calculations in detail. The results in this study mean that we can control the phase states of many osciliators by varying the os¬ cillation frequency of only a few oscillators. Hence, the present study would contribute to the development of coupled oscillators networks [5] which are expected as one of future parallel information processing architec- tures.
Circuit Model
2.
The circuit model is shown in Fig. 2. In the system only the Nth oscillator has different oscillation frequency. vk is the voltage across the capacitor C and ik is the cur- rent through the inductor L of kth oscillator. At first, we assume that v — i characteristics of the nonlinear re- sistor in each oscillator is represented by the following third order polynomial equation. ir (vk) = −g1vk + g3v3
The circuit equation is given as follows. Changing the variables
and defining then the normalized equations are represented as fol- lows
where β is the coupling factor and ε is the largeness of
( V 2 ) I Fig. 2 N oscillators coupled by a resistor with one different frequency oscillator. nonlinearity. Eqs. (7)-(9) may be combined to give the second order nonliner differential equation as follows
where

3.
Circuit Experiments and Numerical Calculations
We carried out circuit experiments and numerical cal- culations for the case of N = 3 and N = 4. For com- puter calculations Eqs. (7)-(9) are calculated by using the Runge-Kutta method with step size Ar = 0.125. Figure 3 shows observed phenomena for N = 3. For the case that all of these oscillators have the same (1) Three-phase synchronization for w = 1.00. (2) Opposite-phase synchronization and oscillation death for ω = 1.64. (3) Opposite-phase synchronization and independent oscillations for ω = 4.47.
frequency (Fig. 3 (1)), we can observe that the system is synchronized at t he three-phase. This phenomenon had been confirmed and analyzed theoretically in [2]. When the frequency of the 3rd oscillator is varied, we ob- served that oscillation of the 3rd oscillator stops in some range of the frequency, namely oscillation death ap- pears. And the others are synchronized at t he opposite- phase (Fig. 3 (2)).  As increasing the frequency of the 3rd oscillator, oscillation of the 3rd oscillator starts again (Fig. 3 (3)). However, the 3rd oscillator is not synchro- nized to the others. Namely, the 3rd oscillator oscillates alone and the others keep opposite-phase synchroniza- tion. Figure 4 shows observed phenomena for N = 4. We can observe similar synchronization phenomena. When the frequencies of the four oscillators are the same, the system is synchronized at t he opposite-phase in pairs and the phase shift between two pairs is not de- cided (Fig.4(1)). Similar phenomenon had been con- firmed in [3], [4]. When the frequency of the 4th os- cillator is varied, oscillation of the 4th oscillator stops, namely oscillation death appears. And the others are synchronized at t he three-phase (Fig.4 (2)). As increas- ing the frequency of the 4th oscillator, the 4th oscillator oscillates alone and the others are synchronized at the three-phase (Fig.4 (3)). Moreover, we observed interesting double-mode os- cillation in the resonance region of w ~ 1 for both case of N = 3 and 4. We will explain this phenomenon later.
Theoretical Analysis
4.
In this section we analyze the synchronization phe- nomena in the previous section by using the averaging method. Equation (12) is first order linear differential equa- tion. The solution is given as follows
In the steady state, the second term of Eq. (14) becomes to zero.  Let us assume the solutions of Eqs. (10)-(11) are
We pay attention to treat the non-resonance system and apply for the averaging method to Eqs.(10)-(11). We
(21) (a) Computer culculation for w = 1.03. (b) Computer culculation for w = 1.05. (c) Computer culculatin for ω = 1.09. (d) Circuit experimental result for L= 10[mH], C=74[nF] and R = 5.6[km]. the stability problem of the solutions in Eqs. (22) (25) is equal to [1], [2]. While the stability of Eqs. (23) (26) are easily confirmed by substituting Eqs. (23) (26) into the equation given by partial differential Eq. (18) with respect to ρ ν .
Resonance Region
ら.
In this section we consider the resonance region, namely ω ~ 1. Experimental and computer calculator results
must be satisfied regardless of the value of β. In addi- tion, if p N is not zero, b N cannot be zero. This means that Nth oscillator is not synchronized to the others. We obtain the solutions as follows ● For the case of N = 3
a For the case of N
(27) In this system when N is more than 4 we can't ob- serve the synchronization without increasing the nonlin- earlty c. If the nonlinearlty is increased, we can't treat the system using the averaging method. For the case of N = 3, Eq. (22) shows that the os- cillators 1 and 2 are synchronized at the opposite-phase. Eq. (23) shows that the amplitude of the 3rd oscillator is a function of w and becomes zero for some range of ω. Eq.(24) shows that the 3rd oscillator does not synchronize to the others. For the case of N = 4, Eq. (25) shows that the os- cillators 1,2 and 3 are synchronized at the three-phase. Eq. (26) shows that the amplitude of the 4th oscillator is a function of w and becomes zero for some range of w. Eq.(27) shows that the 4th oscillator does not synchronize to the others. Using Eq. (23) the relationship between the ampli- tude of the Nth oscillator ρN and the frequency ratio w is shown in Fig. 5. In the figure, we show the numer- ical results together. We can see both results agree well except for resonance region. For the non-resonance case, we can consider that this system is completely separated into the oscillator with the different frequency and the others. Therefor, (25)
(24) (22) (β = 0.1, ε = 0.03), (a) Computer culculation for w = 1.03. (b) Computer culculation for w = 1.05. (c) Computer culculatin for w = 1.09. (d) Circuit experimental result for L=10[mH], C=74[nF] and R = 5.6[kS] for N = 3 are shown in Fig. 6. From the figures we can observe the generation of interesting double-mode os¬ cillation. Namely, oscillators 1 and 2 generate double-mode oscillation and its envelopes are synchronized at the opposite-phase.  While the oscillation of the 3rd os- cillator seems to have only one oscillation frequency, if we neglect small effect. As ω increases, we can observe that the beat frequency of the double-mode oscillation becomes larger and that the amplitude of the 3rd oscil- lator becomes small and vanishes. The observed results for N = 4 are shown in Fig. 7. We can see that the envelopes of the double-mode oscil- lation are synchronized at the three-phase. The changes of beat frequency and the amplitude of 4th oscillator are also confirmed from the figure. Because the averaging method cannot be applied to the double-mode oscillation, we cannot analyze these phenomena theoretically.  However, we would like to emphasize that these phenomena are observed from both of circuit experiments and computer calculations.
Conclusions
6.
In this study, some oscillators with different oscillation frequencies, N — 1 oscillators have the same oscillation frequency and only the N th oscillator has different fre- quency, coupled by a resistor have been investigated. At first we consider non-resonance.  By carrying out circuit experiments and computer calculations, we ob- served that oscillation of the N th oscillator stops in some range of the frequency ratio and that others are synchronized as if the Nth oscillator does not exist. These phenomena are also analyzed theoretically by us- ing the averaging method. Secondly, we investigate the resonance region where the frequency ratio is nearly equal to 1. For this region we can observe interesting double-mode oscillation, that is, synchronization of en¬ velopes of the double-mode oscillation and change of oscillation amplitude of the Nth oscillator. This work partly supported by the Sasakawa Scientific Research Grant from Japan Science Society. [1] 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. [2] H. Kimura and K. Mano, "Three-phase oscillators coupled by resistances," IECE Technical Report, 1965. [3] 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, 1978. [4] Y. Nishio and S. Mori, "Mutually coupled oscillators with an extremely large number of steady states," Proc. of IS- CAS 92, pp.819-822, May 1992. [5] Y. Nishio, S. Mori and A. Ushida "On coupled oscilla- tors networks-For the cellular neural network," Proc. of ISCAS'93, pp.2327–2330, May 1993. Yoshifumi  Nishio received the B.E. and M.E. and Ph.D. degrees in Electrical Engineering from Keio University, Yoko- hama, Japan, in 1988, 1990 and 1993, re- spectively. In 1993, he joined the Depart- ment of Electrical and Electronic Engi- neering at Tokushima University, Toku- shima Japan, where he is currently an Assistant Professor. His research inter- ests are in chaos and synchronization phe- nomena in nonlinear circuits. Dr. Nishio Akio  Ushida received the B.E. and M.E. degrees in electrical engineering from Tokushima University in 1961 and 1966, respectively, and the Ph.D. degree in electrical engineering from University of Osaka Prefecture in 1974. He was an associate professor from 1973 to 1980 at Tokushima University. Since 1980 he has been a Professor in the Department of Electrical Engineering at the university. From 1974 to 1975 he spent one year as a visiting scholar at the Department of Electrical Engineering and Computer Sciences at the University of California, Berke- ley. His current research interests include numerical methods and computer-aided analysis of nonlinear systems. Dr. Ushida is a member of the IEEE.