On the Melnikov method for fractional-order systems

The paper identifies a universal omission in past fractional Melnikov analyses: if the Caputo (or RL/GL) element is defined on a finite memory interval [a,t], the Melnikov perturbation misses portions of the stable/unstable manifolds; to ensure uniform validity one must define memory from −∞ (or +∞ for right derivatives). This is stated explicitly via the double-integral domain in Eq. (4) and the consequence illustrated by Eq. (5), which drops the contribution for t<a (e.g., t<0) and thus neglects W^u on (−∞,0) . The candidate’s Problem A reaches the same conclusion and remedy. For the Duffing–Rayleigh example, the paper derives a closed-form Melnikov function and threshold (Eqs. (22)–(24)), combining standard Rayleigh terms with a fractional term that contains Γ(·)ζ(·) and a trigonometric factor (written there with sec) and a forcing gain with a csch factor . The candidate’s Problem B yields the same structure: K(ω)∝csch and D= Dμ + Dβ + Dγ with Γ(q+2)ζ(q+1) and a trigonometric factor (written with sin), which is consistent up to normalization/representation; the paper’s sec(πq/2) and the model’s sin(πq/2) forms are convertible under the usual Fourier-symbol conventions and gamma-function identities. Minor issues: the model’s explanatory line that writes the “omitted-region term” as −∫0∞… when a=0 mixes up what is omitted versus what is actually retained, but the qualitative claim (omission of (−∞,0)) matches the paper’s Eq. (5) and discussion . Overall: same conclusions; the paper uses complex-analysis residues to evaluate the fractional integral, whereas the model uses a Fourier/Parseval route, leading to equivalent constants in different guises. The closed-form threshold inequality in the paper (Eq. (24)) aligns with the model’s F> D/K(ω) criterion .