The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters

Let a(t),b(t) be continuous T-periodic functions with ∫0Tb(t)dt=0. We establish one stability criterion for the linear damped oscillator x′′+b(t)x′+a(t)x=0. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillator x′′+b(t)x′+a(t)x+c(t)x2n-1+e(t,x)=0, where n≥2,c(t) is a continuous T-periodic function, e(t,x) is continuous T-periodic in t and dominated by the power x2n in a neighborhood of x=0