Introduction

Systems of coupled oscillators are good models for biological rhythmic oscillation such as human circadian rhythms [1], finger movements [2], animal locomotion [3,4] and so on. The investigators have studied the mechanism of oscillation and phase transitions between distinct oscillatory modes. From the standpoint of bifurcation, the former and the latter correspond to Hopf bifurcation of an equilibrium point and D-type of branching of a periodic solution, respectively.

Using group theory, it has been possible to derive some general theorems concerning with the bifurcations of equilibrium points[5,6]. Papy et al. classified equilibrium points and periodic solutions observed in hybridly coupled two oscillators according to their symmetrical properties[7,8]. The equilibrium points are completely classified, however the classification of the periodic solutions is not enough, because they treated the periodic solutions with different symmetrical properties as same type. We think that two coupled oscillators' case is a prototype of modeling to understand the phenomena in a large number of coupled oscillators [9], especially even number of coupled oscillators. By obtaining bifurcation diagrams of a system of coupled oscillators we can design the system with the optimal operating condition.

In this paper we investigate bifurcations of equilibrium points and periodic solutions observed in resistively coupled two oscillators with voltage ports. At first we introduce the definition of a symmetrical equation, equilibrium point and periodic solution [10]. Next we classify the periodic solutions according to their symmetrical properties. By calculating bifurcation sets, transitions between the solutions with different symmetrical properties are obtained. Moreover we find chaotic oscillation created by a cascade of period-doubling bifurcations. As far as we know chaotic oscillation is never reported in such a simple coupled system.