This chapter provides a brief presentation of the so-called Morse-Novikov complex associated with a closed manifold, equipped with a closed non-exact one-form \(\alpha \) and a descending gradient. We are interested in the homoclinic bifurcation of this gradient when crossing the stratum of gradients which have a simple homoclinic orbit, based at a non-degenerate zero of \(\alpha \) . The effect of such a crossing on the Morse-Novikov complex is analyzed. Moreover, an original construction, described in this chapter, shows that the phenomenon of homoclinic bifurcation is very easy to make up. Finally, a surprising doubling phenomenon is associated to every such a bifurcation.