A Condition for Almost Global Stability and Local Asymptotical Stability of Nonlinear System

For nonlinear autonomous systems with the origin as a fixed point, the existence of a density function with its time derivative greater than zero implies almost global stability and local asymptotical stability, which is convenient for verification. Further, if the divergence at the origin is less than zero, the existence of a density function with its time derivative greater than zero is both necessary and sufficient condition for almost global stability and local asymptotical stability. Examples illustrate the validity of the results.