TITLE:
Asymptotic Analysis of Periodic Solutions in Liénard-Type Dynamic Systems
AUTHORS:
Borys Basok, Volodymyr Vladimirovich Gotsulenko
KEYWORDS:
Nonlinear Oscillations, Liénard-Type Dynamic Systems, Limit Cycle, Lindstedt-Poincaré Method, Asymptotic Analysis, Self-Oscillations
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.1,
January
22,
2026
ABSTRACT: This paper investigates the existence, stability, and asymptotic properties of periodic solutions in nonlinear dynamical systems of Liénard type arising in thermohydro-gas-dynamic models. Starting from a physical formulation describing self-oscillations in fluid and gas flows, the governing equations are reduced to a two-dimensional system with a nonlinear characteristic function. Using the Lindstedt-Poincaré method, we construct an asymptotic expansion of the periodic solution in powers of the small parameter, which characterizes the nonlinearity of the system. The approach eliminates secular terms, ensuring uniform validity of the solution over time. Explicit formulas for the amplitude and frequency corrections are derived, and conditions for the existence and stability of the limit cycle are established based on Andronov’s theorem. The results provide analytical insight into the mechanisms of self-oscillation in engineering applications such as compressors and thermoacoustic systems.