Synchronization pattern recognition method for coupled oscillator networks on symmetric graphs based on rotating periodic solutions

Synchronization is an important dynamic behavior in coupled oscillator networks, especially in symmetric networks, where the synchronization characteristics of the system are closely related to the symmetry of its coupling topology. This paper investigates a class of structurally symmetric undirected coupled oscillator network models and proposes a new method for identifying synchronization patterns based on the theory of rotating periodic solutions combined with Laplacian matrix eigenvalue analysis. We find that there is a clear correspondence between synchronization types and eigenvalues of the Laplacian matrix, and the corresponding eigenvectors effectively characterize different synchronization modes. In particular, the appearance of repeated eigenvalues in the Laplacian matrix is a necessary condition for the formation of periodic synchronization and synchronous multistability. This study systematically reveals the relationship between the spectral properties of the Laplacian matrix (including the multiplicity of eigenvalues and the structure of their eigenspaces) and synchronization types. The findings provide new theoretical tools for the identification and prediction of synchronization patterns in complex networks and deepen our understanding of the intrinsic relationship between network structure and synchronization types. Graphical abstract