Dynamics analysis of the fractional-order Lagrange system

The paper’s Propositions 3.1–3.3 compute the Jacobians at e0, e1^m, e2^m, e3^m for the controlled Caputo system (3.3), use Matignon’s criterion |arg(λ)| > qπ/2, and, since all spectra are real, reduce stability to strict negativity of all eigenvalues; this yields exactly: (i) e0 stable iff c1<0 and c2<0; (ii) e1^m stable iff c1<0, c2<0, and |m|<−c2; (iii) e2^m and e3^m stable iff c1<0, c2<0, and |m|<√(c1 c2), with instability otherwise. These match the candidate solution’s conditions and method step-by-step. Minor issues in the paper include a sign typo in the proof of Proposition 3.2, but the stated results are correct. See Proposition 3.1 (e0) , Proposition 3.2 (e1^m) , and Proposition 3.3 (e2^m, e3^m) ; the Jacobians and characteristic polynomials used are as in the paper’s Section 3 , .