Increasing delay as a strategy to prove stability

The paper’s Theorem 4.1 asserts that for a sufficiently smooth map F0 with a hyperbolic fixed point x̄, local asymptotic stability (LAS) is equivalent to the existence of some expansion level m with ||∇Fm||1 < 1; the proof reduces the nonlinear case to the linear one and identifies ||∇Fm||1 with ||(J0^T)^m V0||1, where V0 is the gradient of F0 at x̄ and J0 is the companion-type Jacobian at the equilibrium . In the linear section, the paper develops the expansion machinery (Jm structure, spectra relations), and its main criterion (||pm||ℓ1 < 2) is equivalent to ||Vm||1 < 1 since ||pm||ℓ1 = 1 + ||Vm||1, yielding the same local-stability test in terms of the first row/gradient norm . The candidate solution proves the same equivalence via a direct spectral argument: it establishes ∇Fm = e1^T J^{m+1} (chain rule for the lifted map), then uses the subdiagonal “shift” structure of the companion matrix to show that ||∇Fm||1 < 1 forces all eigenvalues inside the unit disk, and conversely uses J^n → 0 when ρ(J) < 1 to ensure some finite m has ||∇Fm||1 < 1. This is a correct, more explicit proof than the paper’s concise reduction. Hence both are correct; their approaches differ (paper: reduction to linear p_m/row-sum norm via Theorem 3.6; model: direct matrix-power/gradient-row proof) .