ON POLYNOMIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON A CIRCLE WITHOUT SINGULAR POINTS

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.

ANALYSIS USING ELECTROLYTIC LAYER STRIPPING TO DETERMINE THE PARAMETERS OF GRAIN-BOUNDARY DIFFUSION OF COBALT IN POLYCRYSTALLINE NICKEL

A technique for layer-by-layer radiometric analysis using electrolytic layer stripping to determine the parameters of grain-boundary diffusion of cobalt in polycrystalline nickel has been developed. The aim of the work is to develop a complete technological cycle of layer-by-layer radiometric analysis, to select the optimal electrolyte composition for nickel and the conditions for conducting an experiment to remove metal layers with their thickness of 20–200 nm. The studies have been carried out on nickel of nominal purity of 99,98 %. The stabilizing heat treatment of the samples is carried out at a pressure of 10–5 Pa for 2 hours at a temperature of 1273 K. Diffusion annealing is carried out at a pressure of 10–9 Pa in the temperature range of 623–1173 K for 5–30 hours. Concentration profiles have been measured by parallel stripping of layers, which are obtained by electrolytic polishing in a solution based on nickel sulfamate, followed by weighing the sample on a high-precision analytical balance. The difference in weight before and after removal of the layers is used to further calculate the thickness of the removed layers and, as a consequence, the penetration depth of the diffusing. The residual activity of the sample is measured using a digital gamma spectrometer with a NaI (Tl) detector. Before carrying out the experiments, a reference source with a previously known activity has been made to take into account the correction factor due to the radioactive decay of 57Co. The specific layer activity is calculated from the integral remainder of the 122,14 keV line using the Gruzin method. Based on the proposed technique, it is possible to determine the parameters of grain boundary diffusion, such as the diffusion coefficient of grain boundary, triple product and segregation coefficient.