The authors consider the problem of bifurcation of periodic solutions in singular systems of differential equations \[ \varepsilon\dot{u}=f(u)+\varepsilon g(t,u,\varepsilon)\quad u\in\mathbb{R}^n, \] where \(g(t+2,u,\varepsilon)=g(t,u,\varepsilon)\) and \(\dot{u}=f(u)\) has an orbit \(\gamma(t)\) homoclinic to a hyperbolic equilibrium point \(p\). By using a functional analytic approach and the Lyapunov-Schmidt method they obtain a bifurcation function which tends, as \(\varepsilon\to0+\), to the Melnikov function. They show that if a certain Melnikov condition is satisfied then the system has a unique periodic solution of period \(2m\), for any \(m\geq1\), \(m\in\mathbb{N}\), and \(\varepsilon\) sufficiently small.