  A novel method for stability and instability study of autonomous dynamical
systems using the flow and divergence of the vector field is proposed. A
relation between the method of Lyapunov functions and the proposed method is
established. Bendixon and Bendixon-Dulac theorems for $n$th dimensional systems
are extended. Based on the proposed method, the state feedback control law is
designed. The control signal is obtained from the partial differential
inequality. The examples illustrate the application of the proposed method and
the existing ones.
