  We consider a compact manifold of dimension greater than 2 and a differential
form of degree one which is closed but non-exact. This form, viewed as a
multi-valued 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 genericity assumptions these data yield a complex, called today the
Morse-Novikov complex. Due to the non-exactness of the form, its gradient has a
non-trivial dynamics in contrary to gradients of functions. In particular, it
is possible that the gradient has a homoclinic orbit. The one-form being fixed,
we investigate the codimension-one stratum in the space of gradients which is
formed by gradients having one simple homoclinic orbit. Such a stratum S breaks
up into a left and a right part separated by a substratum. The algebraic effect
on the Morse-Novikov complex of crossing S depends on the part, left or right,
which is crossed. The sudden creation of infinitely many new heteroclinic
orbits may happen. Moreover, some gradients with a simple homoclinic orbit are
approached by gradients with a simple homoclinic orbit of double energy. These
two phenomena are linked.
