Four-Dimensional Zero-Hopf Bifurcation for a Lorenz-Haken System

The paper’s main claims (Theorem 1.3) and methodology via first-order averaging are largely consistent and reproduce the advertised bounds on small periodic orbits near zero–Hopf points. However, the paper states an incorrect parameter relation for realizing the zero–Hopf spectrum at the origin: in Proposition 1.2(i) and Theorem 1.3(i) it sets d0 = -√(c^2 + ω^2)/3, whereas the characteristic polynomial calculation requires d0^2 = (c^2 + ω^2)/3, i.e., d0 = -√(c^2 + ω^2)/√3. This inconsistency is visible by comparing the general 3×3 Jacobian cubic at p0 with the target λ(λ^2 + ω^2) and also with the paper’s own formula for the imaginary pair at p (where it correctly gives ±i√(3d^2 - c^2)) and for p± (where it uses the √3 denominator) . Despite this, the subsequent normal-form/averaging computations and stability tests are set up as if the √3 were present (e.g., the repeated appearance of √3 in the change of variables and in the averaged coefficients) . The candidate solution identifies and corrects this error up front and otherwise follows a standard zero–Hopf reduction and averaging argument that agrees with the paper’s results, including the role of the combinations η = 3ce1 + 2√3 d1√(c^2 + ω^2) and η1 = 3a1ω^2 − 2cη in the stability inequalities and the upper bounds 4, 5, and 2 cycles in cases (i)–(iii) respectively . Therefore, with the essential correction d0 = -√(c^2 + ω^2)/√3, the model’s analysis is correct and reconciles the paper’s intended conclusions, while the paper as written contains a substantive parametric error in (i).