A Graphical-Based Method for Plotting Local Bifurcation Diagram

The paper’s algorithm largely matches the exact equilibrium set E = {(x,λ): f(x)+λg(x)=0} and its depiction in the (λ,x)-plane, but key justifications are missing or internally inconsistent. In particular, Step 2’s angle/sign argument contains a parity/sign mistake and contradicts its own statement, and Step 5 is not actually proved from Lemmas 2.1–2.2. By contrast, the candidate solution gives a correct reduction to λ(x) = −f1(x)/g1(x), uses the implicit function theorem near λ=0 at poles, correctly identifies horizontal/vertical asymptotes, and provides a valid stability-labeling rationale for one-dimensional ODEs. Hence, the model is correct; the paper is incomplete.