Hierarchy of partially synchronous states in a ring of coupled identical oscillators

In coupled identical oscillators, complete synchronization has been well formulated; however, partial synchronization still calls for a general theory. In this work, we study the partial synchronization in a ring of N locally coupled identical oscillators. We first establish the correspondence between partially synchronous states and conjugacy classes of subgroups of the dihedral group DN. Then we present a systematic method to identify all partially synchronous dynamics on their synchronous manifolds by reducing a ring of oscillators to short chains with various boundary conditions. We find that partially synchronous states are organized into a hierarchical structure and, along a directed path in the structure, upstream partially synchronous states are less synchronous than downstream ones.

Automated summary

This work investigates partial synchronization in a ring of locally coupled identical oscillators and presents a systematic method to identify all partially synchronous dynamics on their synchronous manifolds. The correspondence between partially synchronous states and conjugacy classes of subgroups of the dihedral group DN is established. The study reveals a hierarchical structure of partially synchronous states, with upstream states being less synchronous than downstream states along a directed path in the structure.