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

## TL;DR

This paper shows how amplitude dynamics in oscillator networks help stabilize synchronization patterns that would otherwise be unstable.

## Contribution

The study introduces nonstructural symmetries to simplify stability analysis and reveals amplitude-phase interactions as crucial for stable synchronization.

## Key findings

- Amplitude dynamics enable linear stability in decoupled oscillator clusters.
- Phase-only oscillators require nonpairwise coupling for similar stability.
- Phase-shift invariance simplifies stability calculations in oscillator networks.

## Abstract

Oscillator networks display intricate synchronization patterns. Determining their stability typically requires incorporating the symmetries of the network coupling. Going beyond analyses that appeal only to a network’s automorphism group, we explore synchronization patterns that emerge from the phase-shift invariance of the dynamical equations and symmetries in the nodes. We show that these nonstructural symmetries simplify stability calculations. We analyze a ring-network of phase–amplitude oscillators that exhibits a “decoupled” state in which physically-coupled nodes appear to act independently due to emergent cancellations in the equations of dynamical evolution. We establish that this state can be linearly stable for a ring of phase–amplitude oscillators, but not for a ring of phase-only oscillators that otherwise require explicit long-range, nonpairwise, or nonphase coupling. In short, amplitude–phase interactions are key to stable synchronization at a distance.