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

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

Jeffrey Emenheiser,^1,2 Anastaysia Salova,^1,2 Jordan Snyder,^1,5 James P. Crutchfield^1,2,3,5, and Raissa M. D'Souza^1,3,4

¹⁾Complexity Sciences Center
²⁾Physics Department
³⁾Computer Science Department
⁴⁾Department of Mechanical and Aerospace Engineering
⁵⁾Mathematics Department
University of California at Davis
Davis, CA 95616

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.

J. Emenheiser, A. Salova, J. Snyder, J. P. Crutchfield, and R. M. D'Souza, “Network and Phase Symmetries Reveal That Amplitude Dynamics Stabilize Decoupled Oscillator Clusters” (2020) .
doi:.
[pdf]
arxiv.org:2010.09131.