Stability and existence of multiperiodic solutions for second-order linear equations with a diagonal differentiation operator

The stability of differential equations with periodic and quasiperiodic coefficients is a central topic in modern stability theory, with important applications in mechanics, physics, and dynamical systems. A classical result in this area is the Lyapunov integral criterion, which provides stability conditions for linear second-order equations with periodic coefficients. In this paper, we extend this criterion to equations with quasiperiodic coefficients. Our analysis is based on the method of periodic characteristics, which has proven effective in the study of multiperiodic solutions for systems with a diagonal differentiation operator. Within this framework, the multiperiodicity condition is reduced to a functional equation, and a Floquet-type representation of the matricant of the associated system is derived. This representation shows that multiperiodicity of solutions follows from the purely imaginary nature of the characteristic multipliers and the periodicity of the helical characteristics. The obtained results confirm that the Lyapunov integral criterion remains valid for equations with quasiperiodic coefficients. More generally, they demonstrate the effectiveness of the characteristic method for analyzing stability in complex dynamical systems, thereby extending the scope of classical stability theory.