On the interpretation of Dirac δ pulses in differential equations for phase oscillators

Both the paper and the model derive the same jump map for the canonically interpreted delta-driven phase ODE: φ^+ = F^{-1}(F(φ^-) + 1), with F(φ) = ∫ dx/Z(x) on intervals where Z does not change sign. The paper presents this via a change of variables and integration across the impulse, yielding equation (12) and the induced modified PRC Z̃(φ) = F^{-1}(F(φ) + 1) − φ (equation (18)), and it recommends using an impulsive formulation with left-limit evaluation dφ/dt = ω + Z(φ^-)∑ δ to realize the intended jump φ^+ = φ^- + Z(φ^-) (equations (21)–(24)) . The model’s solution reproduces the same F-transform derivation (including the fact that Z(φ^-)=0 implies continuity at the impulse) and arrives at the same conclusion; it further supplies a rigorous mollifier-based justification and a higher-order small-PRC expansion, fully consistent with the paper’s first-order Taylor argument (equation (19)) . Hence, both are correct and essentially use the same proof strategy.