On the Synchronization of Oscillators Coupled by One Negative Resistor

There have been many investigations of the mutual synchronization of oscillators. In this study we consider N oscillators which have the same natural frequency, mutually coupled by one negative resistor. In this system, according to the negative range of the coupling negative resistor, we can observe various interesting synchronization phenomena. Because this system tends to minimize the power consumption in the coupling negative resistor. Especially when i v characteristics of the coupling negative resistor are not third-power, but linear and negative, this system tends to be stable by producing the power in the coupling one. The phase states are changed accordeing to the negative range of the coupling negative resistor. INTRODUCTION There have been many investigations of the mutual synchronization of oscillators( [I]-[7] and therein). Kimura and Mano have confirmed that two oscillators coupled by one positive resistor are synchronized at opposite phase, and that the system tends to minimize the power consumption in the coupling positive resistor[l]. Endo et al have analyzed a large number of coupled van der Pol oscillators [2]-[4], and we have confirmed that N-phase oscillation ( N = 2 13) can be stably excited for the case that the nonlinearity of each oscillator is strong [5][6]. In such systems various interesting synchronization phenomena are observed. Because the system tends to minimize the power consumption in the coupling positive resistor, i.e., the system tends to minimize the current through the coupling one, and the sum of output voltage (21 t ~ ) . For the case with third-power nonlinear characteristics, all oscillators exhibit the oscillation. And N-phase oscillation can be stably excited, because of taking any phase differences among 4 = 2k?r/N(lc = 1 , 2 , . . . , N 1). For the case that N is not a prime number, N-phase oscillation seems to be difficult to observe and another types of synchronization are observed. In this study, we investigate synchronization phenom$ Department of Electrical and Electronic Engineering, Tokushima University 2-1 Minami-Jousanjima, Tokushima, 770 JAPAN Email: [email protected]