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.

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The temperature dependences of acoustic properties of nanostructured and polycrystalline zirconium are investigated in the temperature range of 100–340 K. The effect of severe plastic deformation and subsequent annealing on key parameters of the Koiwa–Hasiguti acoustic relaxation in zirconium is studied in detail. It is established that, due to intensive plastic deformation, the relaxation strength considerably increases, and the temperature and the width of the corresponding relaxation peak systematically decrease with reduction of the mean grain size in the samples. Annealing leads to a partial recovery of the relaxation strength and the peak temperature back to the initial values in undeformed samples, but the width of the relaxation peak shows an additional decrease. The majority of the effects observed can be explained by changes in dislocation subsystems of the samples during intensive plastic deformation and annealing. An influence of a random scatter of the relaxation time on the main parameters of the Koiwa–Hasiguti peak is established using the statistical analysis based on the lognormal distribution. It is shown that the parameter β of the lognormal distribution determines the width, height, and asymmetry of the peak and also allows estimating the relaxation strength from the peak height. An algorithm for retrieving the parameter β from experimental data is presented.

The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation we reduce the studied problem to additive eigenvalue problem for a singularly perturbed Hamilton-Jacobi equation. Then assuming that the Aubry set of the Hamiltonian consists of a finite number of points or limit cycles situated in the domain or on its boundary, we find the limit of the eigenvalue and formulate the selection criterium that allows us to choose a solution of the limit Hamilton-Jacobi equation which gives the logarithmic asymptotics of the principal eigenfunction.

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