On the continuum spectra of the exponents of linear homogeneous differential systems

A. Kh. Stash

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 141, pp. 60-67

Résumé

The research subject of this work is at the junction of two sections of the qualitative theory of differential equations, namely: the theory of Lyapunov exponents and the theory of oscillation. In this paper, we study the spectra (i. e., sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential systems with coefficients continuous on the positive semiaxis. For any $n\ge 2$, the existence of an $n$-dimensional differential system with continuum spectra of the oscillation exponents is established. For even $n$, the spectra of all the oscillation exponents fill the same segment of the numerical axis with predetermined arbitrary positive incommensurable ends, and for odd $n$, zero is added to the indicated spectra. It turns out that for each solution of the constructed differential system, all the oscillation exponents coincide with each other. When proving the results of this work, the cases of even and odd $n$ are considered separately. The results obtained are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential systems.

Détail

Keywords: differential equations, linear homogeneous differential systems, spectrum of the system exponent, number of zeros of solution, exponents of oscillation, Sergeev's frequencies.
Mots-clés : oscillation of solution