Stability and Bifurcation Analysis of Two-Term Fractional Differential Equation with Delay

The paper’s Theorem V.1 proves that the characteristic equation λ^α + c λ^{2α} = a + b e^{-λ τ} admits a positive real root (hence instability) for all τ ≥ 0 under the four parameter regimes (i)–(iv), via a P(λ)–Q(λ,τ) intersection argument on the real axis. The candidate solution establishes the same conclusion for the same regimes using a direct intermediate value theorem/sign-change analysis of F_τ(λ) = λ^α + c λ^{2α} − a − b e^{−λ τ}, plus a simple extremum bound when c < 0. Both arguments are sound and reach the same result; they differ mainly in presentation (graphical P–Q vs. IVT). See the paper’s characteristic equation setup and P–Q framework and Theorem V.1 listing conditions (i)–(iv) that ensure instability for all delays .