Properties of Sergeev oscilation characteristics of periodic second-order equation

In this paper, we study the properties of the Sergeev oscillation characteristics of solutions of linear homogeneous second-order differential equations with continuous periodic coefficients. It is known that the upper (weak and strong) oscillation of zeros, roots, hyperroots, strict and non-strict sign changes coincide with the upper Sergeyev frequencies of zeros, roots, and strict sign changes. A similar property holds for all of the listed lower characteristics of Sergeev's oscillation. However, the upper characteristics of solutions of linear homogeneous second-order differential equations with bounded coefficients do not always coincide with the lower ones. In the present paper, equality is established between all characteristics of Sergeev's oscillation on the set of solutions of the Hill equation. Moreover, we have found an effective formula that allows us to find them and conduct studies on the stability of the Hill equation. Besides, a formula connecting Hill equation multipliers with non-integer Sergeyev's frequencies is obtained. Necessary and sufficient conditions of the stability of frequency of the Hill's equation are derived. In proving the results, the transition from Cartesian coordinates to polar coordinates was carried out, so that for the polar angle we obtain an equation that can be interpreted as an equation on the torus. As an auxiliary result, equality was established between the rotation number and the frequency of the Hill equation.