  The current article studies certain problems related to complex cycles of
holomorphic foliations with singularities in the complex plane. We focus on the
case when polynomial differential one-form gives 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. Whenever the polynomial
foliation comes from a perturbation of an exact one-form, one can introduce the
notion of a multi-fold cycle. This type of cycle has at least one
representative that determines a free homotopy class of loops in an open
fibered subdomain of the complex plane. The topology of this subdomain is
closely related to the exact one-form mentioned earlier. We introduce and study
the notion of multi-fold cycles of a close-to-integrable polynomial foliation.
We also explore how these cycles correspond to periodic orbits of a certain
Poincar\'e map associated with the foliation. Finally, we discuss the tendency
of a continuous family of multi-fold limit cycles to escape from certain large
open domains in the complex plane as the foliation converges to its integrable
part.
