Stability of N-front and N-back solutions in the Barkley model

The paper states and proves (by invoking Sandstede’s general theorem after verifying its hypotheses) that, for the Evans–pencil eigenvalue problem Y' = (DXF(γ_{f,N}(ρ)(ξ), ω_N(ρ), r) + λ B(ξ))Y built along the N-front, there is a δ>0 independent of N such that exactly 2N+1 eigenvalues lie in U_δ(0), with asymptotics λ_{2k−1} = (a_{2k−1}+o(1))ρ and λ_{2k} = (a_{2k}+o(1))ρ^{β_ε^2+η_k}, and λ_{2N+1}=0; moreover sign(a_i) = sign(M_f) or sign(M_b), and since M_f,M_b<0, all small eigenvalues lie in the left half-plane. These statements are explicit in Theorem 3.15 together with the setup (176)–(179) and negativity result (190) for the Melnikov integrals, and the recursion defining η_k (171)–(172) . The candidate solution reconstructs the same structure using the standard Evans–Lin reduction: it factors the Evans function via a lower block-triangular Lin matrix, applies Rouché and the implicit function theorem, and concludes the same counting and asymptotics, as well as δ independent of N. This is essentially the method underlying the cited Sandstede theory. One minor difference is that the model explicitly asserts the small eigenvalues are real and negative; the paper only asserts they lie in the left half-plane (consistent with the sign information), but given the real leading-order coefficients this is a benign strengthening rather than a contradiction.