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 degene rate mode in rings of three and four coupled van der Pol oscillators. By us ing the averaging method, we elucidate that the element value deviation has different effects on the rings of three and four coupled oscillators. Name ly, for the ring of three coupled oscillators, two single modes which bifur cate from the twofold degenerate mode in the symmetric system can be synchr onized when they are close, but they lose synchronization and a quasiperiod ic oscillation appears when they are separated to some extent. In contrast, for the ring of four coupled oscillators, the two single modes cannot be s ynchronized, even if they are close. However, in some rare cases they can b e synchronized when they are separated to some extent. We confirm these res ults of the averaging method by both computer simulation and experiments.