Lyapunov reducibility on exceptional rays (Q5951329)

The Lyapunov reducibility problem for the system of linear differential equations \(\dot X=P(t)X\) is posed by \textit{B. M. Levitan} [Almost periodic functions, Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit. (1953)]. It has been proved earlier by the author [Differ. Uravn. 25, No. 6, 1073-1075 (1989; Zbl 0705.34063)] that a special linear system on \(\mathbb{R}^2\) of the type \(\dot X=(P_0+\varepsilon P(q(t))) X\) is Lyapunov reducible, for all sufficiently small values of \(\varepsilon\) possibly except for finitely many rays. In the present paper, it is shown that the cases of Lyapunov reducibility and irreducibility alternate on the exceptional rays.