Group synchrony, parameter mismatches, and intragroup connections

Group synchronization arises when two or more synchronization patterns coexist in a network formed of oscillators of different types, with the systems in each group synchronizing on the same time-evolution, but systems in different groups synchronizing on distinct time-evolutions. Group synchronization has been observed and characterized when the systems in each group are identical and the couplings between the systems satisfy specific conditions. By relaxing these constraints and allowing them to be satisfied in an approximate rather than exact way, we observe that stable group synchronization may still occur in the presence of small deviations of the parameters of the individual systems and of the couplings from their nominal values. We analyze this case and provide necessary and sufficient conditions for stability through a master stability function approach, {which also allows us to quantify the synchronization error}. We also investigate the stability of group synchronization in the presence of intra group connections and for this case, extend some of the existing results in the literature. Our analysis points out a broader class of matrices describing intra group connections for which the stability problem can be reduced in a low-dimensional form.