Global Centers and Phase Portraits in Generalized Duffing Oscillators: A Comprehensive Study of the Center-Focus Problem

The paper’s main characterization of global centers (Theorem C) and its supporting Proposition 4.1 exclude the degenerate case σ=0 for m>1 odd, yet the Hamiltonian system xdot=y, ydot=−x^3 (α=0, ε>0, σ=0, m=3) is a standard counterexample: all regular energy levels H=c>0 are compact ovals filling R^2\{(0,0)}, so the origin is a global center. The paper’s proof of Proposition 4.1 relies on the Poincaré–Lyapunov theorem (linear-center setting) and incorrectly concludes “center iff α=0 and σ>0” for m>1, thereby excluding nilpotent/degenerate centers; Theorem C then inherits this overly strict σ>0 condition. The model’s solution correctly classifies: α=0 and either (i) m=1 with ε+σ>0, or (ii) m>1 odd with ε>0 and σ≥0.