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 integral(T)(0) b(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)x(2n-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 x(2n) in a neighborhood of x = 0.