Hopf bifurcation of a financial dynamical system with delay

At P0, the paper factorizes the characteristic determinant into a scalar delayed factor and a delay-free cubic, namely [λ − (−b + K − K e^{−λτ})](λ^3 + p1 λ^2 + p2 λ + p3) = 0, with p1, p2, p3 given (eqs. (25)–(26)) . It then shows that the cubic is Hurwitz under p1 > 0, p3 > 0, p1 p2 > p3 (eq. (28)) and analyzes the scalar factor, obtaining ω+ = √(2Kb − b^2), the critical delays τj, and the transversality dRe λ/dτ > 0 at τ0 (eqs. (33)–(38)) . The candidate’s Part A mirrors this exactly (rank-one update leading to the same factorization; same ω+, τj, and a closed-form derivative that is algebraically equivalent to eq. (38)). Minor note: re-deriving the 3×3 cofactor at P0 yields p2 = ac + ak + ck + 1 − (c + k − d)/b, while the printed (26b) lists ck + ak + ac − k + c − d/b + 1; the two are not identical, and the former matches the direct cofactor computation (a likely misprint in (26b)). The paper’s conclusions for P0 (Theorem 1) remain unaffected because only the Hurwitz inequalities are used in the argument . At P1, the paper derives a quasi-polynomial characteristic equation R(λ) + Q(λ) e^{−λτ} = 0 with explicit coefficients (eqs. (45)–(46)) and the associated even quartic h(z) in z = ω^2 (eq. (50)) . It then follows Li–Wei’s method to compute candidate imaginary roots and critical delays via an arccos formula (referenced as (60), (62)), imposes the fourth-order Routh–Hurwitz test at τ = 0 (eqs. (63)–(64)), and proves transversality via an explicit derivative (eq. (80)), concluding a Hopf bifurcation at τ0 (Theorem 2) . The candidate’s Part B is the same strategy: same structure R(λ)+Q(λ)e^{−λτ}=0, the same quartic h(z) with p, q, u, v as in (50), identical Routh–Hurwitz conditions at τ = 0 (63)–(64), the same arccos-based critical delay formula, and the same transversality condition via h′(z0) ≠ 0. Coefficient-by-coefficient checks show the candidate’s a1,b2,c2,d1 reduce to the paper’s (46) when θ^2 is eliminated using θ^2 = kb(1+ac)/(c(d−k)) + 1. In short, both analyses reach the same stability/Hopf conclusions at P0 and P1 with substantially the same proof technique, with the candidate adding some algebraic detail and correcting a probable typo in p2 at P0. The paper’s narrative links (Theorem 1/2, Lemmas 3–8) and numerical validations are consistent with the candidate’s claims (figures and computed τ0) .