Fixed-Time Leaderless Cluster Synchronization of Spatiotemporal Community Networks With Coopetition Interactions.

This article addresses the fixed-time leaderless cluster synchronization of spatiotemporal community networks (SCNs) characterized by nonidentical node dynamics and reaction-diffusion feature. First, a signed SCN with reaction-diffusion effect is formulated, where the sign-based coupling is introduced to capture the dynamics of coopetition interactions among different communities. Second, to ensure the invariance of the synchronous manifold, an improved interdegree balance condition is proposed as a prerequisite for achieving cluster synchronization of the community network. Third, based on the local state information from adjacent nodes within each community, a time-limited controller is designed to enhance intracommunity coordination while avoiding the adverse effects of intercommunity competition on synchronization. Subsequently, with the help of the matrix decomposition technique and a Lyapunov-like method, several flexible leaderless cluster synchronization criteria are derived by establishing a nontrivial integral inequality and key properties of the intracommunity Laplacian matrix. Finally, the theoretical results are substantiated through a numerical example.