  This paper proposes a method for certifying the local asymptotic stability of
a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares
(SOS) programming to search for a partially quadratic Lyapunov Function (LF).
The proposed method is particularly well suited to the stability analysis of
ODEs with high dimensional state spaces. This is due to the fact that partially
quadratic LFs are parametrized by fewer decision variables when compared with
general SOS LFs. The main contribution of this paper is using the Center
Manifold Theorem to show that partially quadratic LFs that certify the local
asymptotic stability of a given ODE exist under certain conditions.
