Stability region analysis for polynomial systems based on density function

For nonlinear dynamical systems with the origin as a fixed point, this paper presents an asymptotical stability criterion based on density function. The criterion shows that the existence of a positive density function with its time derivative greater than zero implies local asymptotical stability. The criterion has a remarkable application, which in this note is used for estimating the region of attraction (RoA). Further, we propose using (bilinear) sum-of-squares programming for obtaining inner bounds of RoAs for dynamical systems with polynomial vector fields. Examples illustrate the validity of the results.