This note applies the direct method of Lyapunov to stability analysis of a dynamical system with multiple nonlinearities. The essential feature of the Lyapunov function used in this note is a non-Luré type Lyapunov function which surpasses the Luré-type Lyapunov function from the point of view of the stability region guaranteed. A modified version of the multivariable Popov criterion is used to construct non-Luré type Lyapunov function, which allow for the dynamical sytems with multiple nonlinearities.