An explicit Hopf bifurcation criterion of fractional-order systems with order 1 < α < 2

The paper defines the fractional Routh–Hurwitz matrix H_α from the boundary-evaluated polynomial g(ir)=f(r e^{iαπ/2};α,μ)=f1(r)+i f2(r), and uses its leading principal minors ∇p together with a specific cofactor ∇̃ to give explicit Hopf conditions for 1<α<2. The main criterion requires: (i) eigenvalue condition ∇n(μ*)=0, ∇p(μ*)>0 for p=1,…,n−1, and ∇̃(μ*)<0 (capturing a simple pair on the boundary ray), and (ii) a transversality condition that ∇n(μ) changes sign in every neighborhood of μ*; see Definition 1 and Theorem 1 for H_α, ∇p and ∇̃, and the two-part Hopf test . The proofs hinge on the generalized Routh–Hurwitz criterion in a rotated coordinate system (so the critical line becomes the y′-axis), identification of ∇n with the resultant Res(f1,f2), and a subresultant identity giving the boundary radius r0=−∇̃/∇n−1; see Lemma 1, Theorem 2 and identity (11) derived from subresultants . The candidate model’s solution follows the same structure: reduce to sector root location via λ=r e^{iαπ/2}, identify ∇n as the resultant and ∇̃ as the first non-vanishing subresultant, and then use the generalized Routh–Hurwitz/Sturm machinery to count roots inside the sector and deduce a unique transverse crossing when ∇n changes sign while the lower minors stay positive. This matches the paper’s method and intent; the model adds a standard pivot-sign/zero-count interpretation that the paper references but does not spell out. Overall, both are correct and essentially the same proof skeleton, with the paper presenting it as Theorems 1–3 and the model filling in well-known Routh–Hurwitz counting details.