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The cluster synchronization is a very important characteristic for the higherharmonic coupling Kuramoto system. A novel transformation is provided, and itgives cluster synchronization by the periodic properties of the densityfunction. The periodic properties of the density function also make the clustersections' boundaries barrier-like, which helps to explain the sensitiveness ofcluster synchronization on the initial conditions of the oscillators. Detailednumerical studies confirm the theoretical predictions from this new view of thesymmetry transformation. The work is very beneficial to the further study oncluster synchronization in various systems.

The new explanation of cluster synchronization in the generalized Kuramoto system