Effects of the deviation of element values in a ring of three and four coupled van der Pol oscillators

This paper investigates the effect of element value deviation on the degenerate mode in rings of three and four coupled van der Pol oscillators. By using the averaging method, we elucidate that the element value deviation has different effects on the rings of three and four coupled oscillators. Namely, for the ring of three coupled oscillators, two single modes, which bifurcate from the two-fold degenerate mode in the symmetric system can be synchronized when they are close, but they lose synchronization and a quasi-periodic oscillation appears when they are separated to some extent. In contrast, for the ring of four coupled oscillators, the two single modes cannot be synchronized even if they are close, but in some rare cases they can be synchronized when they are separated to some extent. We confirm these results of the averaging method by both computer simulation and experiments.