  The cluster synchronization is a very important characteristic for the higher
harmonic coupling Kuramoto system. A novel transformation is provided, and it
gives cluster synchronization by the periodic properties of the density
function. The periodic properties of the density function also make the cluster
sections' boundaries barrier-like, which helps to explain the sensitiveness of
cluster synchronization on the initial conditions of the oscillators. Detailed
numerical studies confirm the theoretical predictions from this new view of the
symmetry transformation. The work is very beneficial to the further study on
cluster synchronization in various systems.
