Periodic solutions of a class of second-order non-autonomous differential equations

This paper investigates the dynamical behavior of periodic solutions for a class of second-order non-autonomous differential equations. First, based on the Lyapunov-Schmidt reduction method for finite-dimensional functions, the corresponding bifurcation function is constructed, and it is proven that the system possesses at least one T-periodic solution. Second, a two-timing method is employed to perform perturbation analysis on the original equation. By separating the fast and slow time scales, an explicit expression for the approximate T-periodic solution is derived. Furthermore, for the stability of the system under parametric excitation, the bifurcation characteristics near the first instability tongue are revealed through perturbation expansion and eigenvalue analysis. Additionally, the Ince-Strutt stability diagram is plotted to illustrate the stability boundaries.