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An outstanding problem in the study of networks of heterogeneous dynamicalunits concerns the development of rigorous methods to probe the stability ofsynchronous states when the differences between the units are not small. Here,we address this problem by presenting a generalization of the master stabilityformalism that can be applied to heterogeneous oscillators with largemismatches. Our approach is based on the simultaneous block diagonalization ofthe matrix terms in the variational equation, and it leads to dimensionreduction that simplifies the original equation significantly. This newformalism allows the systematic investigation of scenarios in which theoscillators need to be nonidentical in order to reach an identical state, whereall oscillators are completely synchronized. In the case of networks ofidentically coupled oscillators, this corresponds to breaking the symmetry ofthe system as a means to preserve the symmetry of the dynamical state---arecently discovered effect termed asymmetry-induced synchronization (AISync).Our framework enables us to identify communication delay as a new andpotentially common mechanism giving rise to AISync, which we demonstrate usingnetworks of delay-coupled Stuart-Landau oscillators. The results also havepotential implications for control, as they reveal oscillator heterogeneity asan attribute that may be manipulated to enhance the stability of synchronousstates.

Identical synchronization of nonidentical oscillators: when only birds of different feathers flock together