On a Gradation for Asymptotic Stability

Classical asymptotic stability guarantees convergence but does not quantify the rate at which convergence occurs. This paper introduces a gradation of asymptotic stability where degree zero corresponds to exponential stability and degree $m>0$ corresponds to algebraic decay of order $t^{-1\mathrm{/}m}$. We provide direct and converse Lyapunov tests for admissible degrees and conditions for certifying the exact stability degree. Hopf, Bautin, fractional-degree, and time-varying examples demonstrate how the degree identifies the leading stabilizing mechanism.