Phase-lock area portrait and ω→0 asymptotics for the dRSJ family (open even for RSJ)

Determine the portrait of phase-lock areas in the (B, A)-parameter plane for the two-parameter deformed RSJ (dRSJ) dynamical system family defined by dθ/dτ = (cos θ + B + A sin τ) / (ω(1 − δ cos τ)) + D with fixed ω, δ, and D. Derive the asymptotic behavior of this phase-lock area portrait as ω → 0 for fixed δ ∈ [0,1) and D ∈ R. Note that this remains unresolved even for the classical RSJ model obtained at δ = 0 and D = 0.

Background

The paper introduces the deformed RSJ (dRSJ) family of dynamical systems on the two-torus, given by dθ/dτ = (cos θ + B + A sin τ) / (ω(1 − δ cos τ)) + D, which generalizes the classical RSJ model (recovered at δ = 0). The dRSJ family is connected to general Heun equations via projectivized linear systems, and it exhibits rotation number quantization, while showing a qualitative change in phase-lock geometry (constrictions break).

Phase-lock areas are level sets of the rotation number with non-empty interior in the (B, A)-plane. The authors call for a full description of the global phase-lock portrait for fixed (ω, δ, D), and, importantly, for asymptotic characterization as ω → 0. They explicitly note that this asymptotic problem is open even for the original RSJ model (δ = 0, D = 0), underscoring the challenge and foundational status of the question.