We consider an autonomous perturbation of an autonomous system of ordinary differential equations, when a periodic solution of the autonomous system has a nontrivial multiplier equal to $1$ or $-1$. We derive criteria for the perturbed system to have an orbitally asymptotically stable periodic solution using a technique of stable fixed points of monotone operators together with a bifurcation technique due to M.A. Krasnoselskii.
"Stable periodic solutions of perturbed autonomous equations in one critical case." Differential Integral Equations 8 (5) 1135 - 1143, 1995.