Some results on the asymptotic stability of second-order nonlinear systems

This note gives some stability results concerning second-order systems
\dot{x} = f(x)
, where
f(x)
contains either linear and quadratic or linear and cubic terms in
x
. Following a Lyapunov-like approach, a closed-form estimate for asymptotic stability regions of such systems is derived in terms of quadratic functions and then it is optimized with respect to its area. Some application examples show the good results of the proposed method, in comparison to those obtained by the classical numerical approaches.