Stability analysis of the nonlinear pendulums under stochastic perturbations

The paper correctly derives the averaged potential Ū(θ)=Λ1 cos 2θ+Λ2 sin 2θ−gl cos θ and gives the parametric degeneracy curve Γ1, but then states two flawed items: (i) an incorrect symmetry b) that would allow restricting to Λ1≥0,Λ2≥0, and (ii) the equal-energy-saddles ray Γ2 as {Λ1>0.25,Λ2=0}. For Ū with the minus cosθ term used in the paper, the two equal-energy saddles occur precisely on the line Λ2=0 for Λ1≤−1/4, not Λ1>1/4. The candidate solution derives Γ1 correctly and gives a correct derivation and sign for Γ2 for the potential as written, alongside a consistent averaging estimate. See the paper’s Ū and averaging setup (eq. (4.3)) and its stated bifurcation curves (Γ1,Γ2) as printed, which conflict with the saddle condition for Ū having −cosθ (and the printed symmetry b) is false) . The paper’s probabilistic averaging bounds match the model’s optional Step 3 in spirit, cf. their Ũ(t,θ) and P(|H−H̄|>δ)→0 as σ→0 .