We study the first eigenpair of a Dirichlet spectral problem for singularly perturbed
convection-diffusion operators with oscillating locally periodic coefficients. It follows
from the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularly
perturbed operators with oscillating coefficients. Preprint
www.arxiv.org, arXiv:1206.3754] that the
first eigenvalue remains bounded only if the integral curves of the so-called effective
drift have a nonempty ω-limit set. Here we consider the case when the
integral curves can have both hyperbolic fixed points and hyperbolic limit cycles. One of
the main goals of this work is to determine a fixed point or a limit cycle responsible for
the first eigenpair asymptotics. Here we focus on the case of limit cycles that was left
open in [A. Piatnitski and V. Rybalko, Preprint.