Interval of η for incoherence–coherence transition under subgraph constraint

Determine whether the transition from incoherence to partial synchrony in the adaptive Kuramoto oscillator network with a two-population subgraph constraint—implemented by enforcing decay of inter-population edge weights at rate η^{-1}—is restricted to a specific interval of the homeostasis parameter η. The network consists of N=2Q oscillators with sinusoidal coupling and adaptive weights evolving according to A(φ_k, φ_j, w_{kj}) = ε(b + a cos(φ_j − φ_k + β) − w_{kj}), and the constraint splits the network into two disconnected Q-node populations when η → 0.

Background

The paper studies adaptive Kuramoto oscillator networks where edge adaptation is subject to linear constraints, enforced via a homeostasis term that acts on weights transverse to a constraint manifold with timescale parameter η. One example is a subgraph constraint that partitions N=2Q oscillators into two disconnected populations when η→0, progressively relaxing to global uniform adaptivity as η increases.

In this setup, the authors observe transitions between incoherence and partial synchrony as η varies. They note multistability and the absence of a unique critical η, and pose the explicit question of whether the transition is confined to an interval of η values, highlighting the need for a precise characterization of the transition regime.