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