On the implementation of countable essential spectra of oscillation exponents in a linear homogeneous two-dimensional differential system

Various types of oscillation exponents are studied for solutions of linear homogeneous differential systems with continuous bounded coefficients. The oscillation exponents are calculated by averaging the number of zeros (or signs, or roots, or hyperroots) of the projection of a solution $x$ of a differential system onto some straight line, where this line is chosen so that the resulting average value is minimal. If minimization precedes averaging, then the weak oscillation exponents are obtained; otherwise, the strong ones. When calculating oscillation exponents of a solution $y$ of a linear homogeneous $n$th-order differential equation, we pass to the vector function $x=(y, \dot y,\dots, y^{(n-1)})$. These exponents are closely related to the intersections of solutions with hyperplanes (passing through the origin), i.e. with the oscillation of projections of these solutions onto all possible straight lines. The main result of the paper is an explicit construction of a two-dimensional linear bounded system with the property that its spectra of all upper and lower, strong and weak oscillation exponents of strict and nonstrict signs, zeros, roots, and hyperroots coincide with any predetermined closed bounded countable set of nonnegative rational numbers with a single zero limit point. Moreover, for any nonzero solution of the constructed system, all oscillation exponents coincide, and each of their values is metrically and topologically essential. In the construction of the mentioned system and in the proof of the main result, we apply analytical methods of the qualitative theory of differential equations and a special method of controlling the fundamental matrix of a two-dimensional differential system.