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

Abstract

Let $a\left(t\right),b\left(t\right)$ be continuous $T$-periodic functions with ${\int }_0^{T}b\left(t\right)dt=0$. We establish one stability criterion for the linear damped oscillator $x^{\mathrm{\prime \prime }}+b\left(t\right)x^{\prime }+a\left(t\right)x=0$. 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 $x^{\mathrm{\prime \prime }}+b\left(t\right)x^{\prime }+a\left(t\right)x+c\left(t\right)x^{2n-1}+e(t,x)=0$, where $n\ge 2,c\left(t\right)$ 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 $x=0$.