Conclusions

We investigated bifurcations of equilibrium points and periodic solutions observed in a resistively coupled oscillator with voltage ports. Firstly we classified theoretically all equilibrium points and periodic solutions according to their symmetrical properties. Eight types of periodic solutions including in-phase and anti-phase solutions [14] were shown. Secondly by calculating bifurcation sets numerically we demonstrated the existence of the equilibrium points and the periodic solutions which we classified. Lastly we showed transitions of solutions between different types of symmetries by symmetry-breaking bifurcation. Moreover we found chaotic oscillation created by successive period-doubling bifurcations. As far as we know, this is the first observation of chaotic oscillation in such a simple coupled oscillators system

Extension to large number of oscillators and calculation of global bifurcation sets are interesting problems for the future.