The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters
Summary: Let \(a(t),b(t)\) be continuous \(T\)-periodic functions with \(\int^T_0 b(t)\,dt = 0\). We establish a 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 \(n\geq 2\), \(c\) 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 \(x=0\).
scientific article; zbMATH DE number 5791613