Abstract The combination of the averaging method with the Lyapunov-Schmidt reduction offers sufficient conditions for establishing the existence of periodic solutions in a specific class of perturbative T -periodic nonautonomous differential equations, given by the form x'=F₀ (t, x) + F (t, x, ) x ′ = F 0 (t, x) + ε F (t, x, ε). These periodic solutions emerge from a manifold Z Z of periodic solutions of the unperturbed system x'=F₀ (t, x) x ′ = F 0 (t, x). The stability analysis of these periodic solutions requires computing the eigenvalues of matrix-valued functions M () M (ε), typically involving the theory of k -hyperbolic matrices. Traditionally, the theory demands a diagonalization process for the k -jets of M () M (ε), for which no general algorithm exists. In this study, we propose an alternative strategy to determine the stability of periodic solutions without relying on such diagonalization, making our approach applicable even in cases where diagonalization is not feasible. Additionally, we present applications of our results for two families of 4D vector fields: the first family corresponds to a perturbation of a two-degree-of-freedom Hamiltonian system, while the second family corresponds to a perturbation of a 4D vector field with a one-degree-of-freedom Hamiltonian sub-system.
Cândido et al. (Sat,) studied this question.