A geometric approach to periodically forced dynamical systems in presence of a separatrix

We consider the periodic boundary value problem for the non-autonomous scalar second-order equation ẍ+F(x,ẋ)=e(t), with e(·) a continuous and T-periodic forcing term. Using a continuation theorem adapted from Capietto et al. (Trans. Amer. Math. Soc. 329 (1992) 41-72), we propose some new conditions for the existence of T-periodic solutions to the forced equation in terms of the dynamical properties of the trajectories of the associated autonomous equation ẍ+F(x,ẋ)=0. Special emphasis will be addressed to the study of the case in which the presence of an unbounded separatrix for the autonomous system in the phase-plane allows to obtain a priori bounds for the T-periodic solutions of the homotopic equation ẍ+F(x,ẋ)=λe(t).