  In the current article we study complex cycles of higher multiplicity in a
specific polynomial family of holomorphic foliations in the complex plane. The
family in question is a perturbation of an exact polynomial one-form giving
rise to a foliation by Riemann surfaces. In this setting, a complex cycle is
defined as a nontrivial element of the fundamental group of a leaf from the
foliation. In addition to that, we introduce the notion of a multi-fold cycle
and show that in our example there exists a limit cycle of any multiplicity.
Furthermore, such a cycle gives rise to a one-parameter family of cycles
continuously depending on the perturbation parameter. As the parameter
decreases in absolute value, the cycles from the continuous family escape from
a very large subdomain of the complex plane.
