Symmetric higher-order interactions in Kuramoto oscillators with time-delayed coupling: Emergence of chimera states and multi-cluster synchronization

We investigate the dynamics of Kuramoto oscillators with symmetric higher-order interactions in the presence of time delays. Unlike previous studies focusing on asymmetric interactions, we analyze symmetric coupling terms of the form ∑j,ksin(θj+θk−2θi+ϕ), where ϕ represents a phase lag parameter. Using the Ott–Antonsen (OA) ansatz, we derive reduced equations for the order parameter and demonstrate that time delays can induce novel collective behaviors including chimera states, multi-cluster synchronization, and explosive desynchronization. Our analysis reveals that the interplay between symmetric higher-order interactions and time delays creates bistable regions where coherent and incoherent states coexist. We identify critical delay values that promote the emergence of partial synchronization patterns and show that the phase lag parameter ϕ controls the transition between different synchronization regimes. Numerical simulations with N=104 oscillators confirm our analytical predictions and reveal rich dynamics including traveling wave solutions, spatiotemporal chaos, and novel breathing chimera states. The system exhibits up to five coexisting stable states in certain parameter regions, significantly extending the multistability observed in previous higher-order Kuramoto models.