Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Proposition 5.1 in the paper asserts that the sets R7 and R8 are transcritical bifurcation loci for Family V, but the proof is only a schematic region-chaining argument (E3 → R7 → E2 and E4 → R8 → E1) with inconsistent notation (mixing E- and R-sets), an apparent sign error for P2, and without a precise definition of R7 and R8 or a direct Jacobian-based verification of the exchange-of-stability criterion stated earlier in the paper’s own definition of “transcritical” (namely, two equilibria exist for every parameter value and exchange stability after the collision) . By contrast, the candidate solution computes both equilibria P1=(0,0) and P2=(−3b/(2c),0), evaluates their Jacobians, shows tr(J) is equal at both while det(J) has opposite sign ±(3/2)b, and concludes that as b crosses 0 the saddle/non-saddle roles exchange—in full agreement with the paper’s definition of a transcritical bifurcation. This is further corroborated by a 1D center-manifold reduction at b=0 (with d≠0), yielding the normal form ẋ=μx+νx^2, and by the Hamiltonian case at s=−4 (i.e., d=0), where the divergence vanishes and centers arise as appropriate . The conclusion is that the result is essentially correct, but the paper’s proof is incomplete and contains presentation errors, whereas the model’s proof is correct and complete at the local (Jacobian/center-manifold) level.