RESONANCES IN NONLINEAR SYSTEMS WITH A DECAYING CHIRPED-FREQUENCY EXCITATION AND NOISE

The paper proves stochastic persistence of resonant phase-locking for a chirped, decaying perturbation via amplitude–angle variables, averaging to a resonant normal form, and a stochastic Lyapunov/supermartingale argument that yields three time-horizon regimes depending on C versus A + B/2. The candidate solution follows the same roadmap—same scalings (A,B,C), same averaged coefficients P(Θ), J(Θ), the same phase-locking condition, and the same probability bound with identical time horizons. One technical mismatch is the candidate’s use of η = (A + B/2 − C)/B as the decay exponent for the noise in slow time; the paper’s explicit transformed SDE shows the governing exponent for the quadratic variation is 2(C − A)/B, which gives the correct threshold C ≶ A + B/2. Despite this notational/sign slip, the candidate’s final statements and bounds align with the paper’s results.