Non-uniform asymptotic stability for the damped linear oscillator

Sufficient conditions are established for non-uniform asymptotic stability of a linear oscillator with damping term. The obtained results clarify a difference between the uniform asymptotic stability and the asymptotic stability. Some simple examples are included to illustrate the results. Especially, Bessel’s differential equations are taken up and it is proved that the equilibrium is asymptotically stable, but it is not uniformly asymptotically stable.