Heterogeneous phase oscillators under symmetry breaking of coupling strength: Optimal synchronization and desynchronization under strong gain

Revealing the collective dynamics of heterogeneous oscillator networks under asymmetric coupling is a great challenge. In this study, we discuss optimal synchronization and desynchronization under strong gain in heterogeneous phase oscillator networks with symmetry breaking of coupling strength. Firstly, for a fixed coupling strength, we discuss how the heterogeneity parameter affects the synchronization ability of the system. Secondly, for different values of the heterogeneity parameter, we propose the concepts of gain factor and weakening factor. We find that the synchronization ability of the system is greatly weakened due to the presence of the weakening factor. On the contrary, the presence of the gain factor allows the system to quickly transition to a completely synchronized state even at a small coupling strength. Thirdly, we find an interesting phenomenon that distinguishes the symmetrically coupled heterogeneous phase oscillator networks: the system under symmetry breaking of coupling strength exhibits a significant desynchronization behavior at strong gain. Finally, for a fixed heterogeneity parameter, we give the optimal intrinsic frequency assignment scheme based on the spectral decomposition of the Laplace matrix of the underlying network. Theoretically, these findings may help us to better understand the collective dynamics in heterogeneous phase oscillator networks under asymmetric coupling that are prevalent in the real world. In particular, they provide inspiration and guidance for optimizing the synchronization of heterogeneous phase oscillator networks under symmetry breaking of coupling strength.