Some Extensions of Liapunov's Second Method

>In the study of the stability of a system, it is never completely satisfactory to know only that an equilibrium state is asymptotically stable. As a practical matter, it is necessary to have some idea of the size of the perturbations the system can undergo and still return to the equilibrium state. It is never possible to do this by examining only the linear approximation. The effect of the nonlinearities must be taken into account. Liapunov's second method provides a means of doing this. Mathematical theorems underlying methods for determining the region of asymptotic stability are given, and the methods are illustrated by a number of examples.