Synchronization control of coupled phase oscillators: An analytical framework with a partial freezing scheme

Synchronization control lies at the heart of research on collective behavior in complex systems. By actively regulating the emergent synchronization in coupled oscillator systems, one can gain profound insights into non-equilibrium phase transitions and the principles governing nonlinear spatiotemporal evolution. Building upon the concept of static pinning control in networks, this paper introduces a partial phase-freezing control scheme. Through analytical investigations, we systematically examine the role of frozen oscillators in shaping the synchronization emergence of unfrozen ones. The study focuses on how the proportion of frozen oscillators modulates the emergent properties of synchronization manifolds—specifically, the synchronization performances, synchronization capability, and attractor convergence characteristics. The results reveal that partial phase freezing alone is sufficient to markedly alter, and even reconfigure, the collective dynamics of phase oscillator systems; remarkably, an extremely small fraction of frozen oscillators can induce synchronization states that are inherently unattainable in the original system.