Homoclinic bifurcation in Morse–Novikov theory, a doubling phenomenon

We consider a compact manifold of dimension greater than 2 with a differential form of degree one which is closed but non-exact. This form, viewed as a multi-valued real function has a gradient vector field with respect to any Riemannian metric. After S. Novikov’s work and a complement by J.-C. Sikorav, under some genericness assumptions, this data yields a complex, called today the Morse–Novikov complex. Due to the non-exactness of the form, its gradient may have a homoclinic orbit. The one-form being fixed, we investigate the codimension-one stratum, in the space of gradients, formed by those having only one simple homoclinic orbit. The crossing of such a stratum has a subtle effect on the Morse–Novikov complex: some crossing may creates infinitely many new heteroclinic orbits; and some simple homoclinic orbit may be approached by simple homoclinic orbits of double energy. These latter two phenomena are linked.