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Oscillator networks display intricate synchronization patterns. Determiningtheir stability typically requires incorporating the symmetries of the networkcoupling. Going beyond analyses that appeal only to a network's automorphismgroup, we explore synchronization patterns that emerge from the phase-shiftinvariance of the dynamical equations and symmetries in the nodes. We show thatthese nonstructural symmetries simplify stability calculations. We analyze aring-network of phase-amplitude oscillators that exhibits a "decoupled" statein which physically-coupled nodes appear to act independently due to emergentcancellations in the equations of dynamical evolution. We establish that thisstate can be linearly stable for a ring of phase-amplitude oscillators, but notfor a ring of phase-only oscillators that otherwise require explicitlong-range, nonpairwise, or nonphase coupling. In short, amplitude-phaseinteractions are key to stable synchronization at a distance.

Network and Phase Symmetries Reveal That Amplitude Dynamics Stabilize Decoupled Oscillator Clusters