Dynamics of Low-Dimensional Quadratic Families

The paper’s Theorem 3.2 states exactly the same classification for fixed points of Fc(x)=x^2+c/x with c>0—one repelling fixed point when c>4/27; two fixed points with the positive one neutral when c=4/27; and three fixed points with the middle one attracting when 0<c<4/27—and correctly uses the identity F′c(x0)=3x0−1 at fixed points to assign stability . However, the argument in the paper does not supply a self-contained, rigorous count of real fixed points across parameter ranges; it relies on “observing Fig. 5” and descriptive statements of a saddle-node at c=4/27 without a calculus-based proof of the cubic p(x)=x^3−x^2+c having 1/2/3 real roots as c varies. By contrast, the candidate solution gives a complete calculus proof via p′ and the critical values p(0)=c and p(2/3)=c−4/27, then derives stability from F′c(x0)=3x0−1. Thus the model’s solution is correct and complete, while the paper’s presentation of this particular result is correct but logically incomplete.