A general frequency domain stability criterion is presented for autonomous nonlinear dynamical systems that possess several non-linearities and several equilibrium states. As special cases, the result is shown to contain well-known criteria for non-oscillatory behaviour of non-linear feedback loops, of non-linear electrical RLC-circuits and analogue neural networks. The proof relies on a single Liapunov function which can subsequently be used lo compute regions of attraction for each of the equilibrium states. Therefore the result is particularly useful for the analysis and the design of systems such as neural classification networks, which possess many equilibrium states.