An outstanding problem in the study of networks of heterogeneous dynamical units concerns the development of rigorous methods to probe the stability of synchronous states when the differences between the units are not small. Here, we address this problem by presenting a generalization of the master stability formalism that can be applied to heterogeneous oscillators with large mismatches. Our approach is based on the simultaneous block diagonalization of the matrix terms in the variational equation, and it leads to dimension reduction that simplifies the original equation significantly. This new formalism allows the systematic investigation of scenarios in which the oscillators need to be nonidentical in order to reach an identical state, where all oscillators are completely synchronized. In the case of networks of identically coupled oscillators, this corresponds to breaking the symmetry of the system as a means to preserve the symmetry of the dynamical state—a recently discovered effect termed asymmetry-induced synchronization (AISync). Our framework enables us to identify communication delay as a new and potentially common mechanism giving rise to AISync, which we demonstrate using networks of delay-coupled Stuart–Landau oscillators. The results also have potential implications for control, as they reveal oscillator heterogeneity as an attribute that may be manipulated to enhance the stability of synchronous states.
This movie starts with two systems of identical oscillators, denoted by +h and -h, respectively. Both systems are unstable and eventually become desynchronized. Half way through the video, we changed the off-diagonal systems from homogeneous to heterogeneous, which consists of a mixture of +h and -h oscillators. This heterogeneous system is able to maintain synchronization, either in the form of limit-cycle oscillation or oscillation death.
- Y. Zhang and A. E. Motter, Identical synchronization of nonidentical oscillators: when only birds of different feathers flock together, Nonlinearity 31 R1-R23 (2018)