Comparative Analysis of Monotonically Stable Solutions of Differential Equations Limited by an Asymptote

The article is devoted to the comparative analysis of pairs of monotonically stable solutions of ordinary differential equations having a common asymptote. It is assumed that monotonically stable solutions are continuous, nonlinear, and also have a limited quantity of inflection points. It is known that the application of the classical mathematical analysis makes it possible to obtain conditions of the simultaneous monotonic stability of two nonlinear solutions under study. In the comparative analysis of the monotonically stable solutions in a pair, elements of the analysis of infinitesimal functions are used. The aim of the work is to analyze the qualitatively different cases describing the variation of the pairs of the simultaneously monotonically stable solutions that limited by a common asymptote and located in the upper half-plane of the x-y plane. The article takes into account that the monotonically stable solutions can have one: vertical, inclined or horizontal asymptote. Ultimately, seventeen qualitatively different cases of variation of the pairs of the simultaneously monotonically stable solutions limited by an asymptote in the upper half-plane are identified. In a similar way, one can carry out a comparative analysis of the cases of the simultaneous monotonic stability of the solutions with a common asymptote in the lower half-plane of the x-y plane. #CSOC1120.