This paper presents a new analytical describing-function based method which utilizes the phenomenon of attractors to predict the existence of stable, unstable and strange attractors in autonomous nonlinear systems. This method serves as a criterion for a route to chaos, which is an extension to the traditional criteria for limit cycle stability, thus bridging a gap between two apparently separate worlds in dissipative systems theory, namely traditional harmonic linearization techniques and the relatively new theory of strange attractors. An illustration of the procedure is presented by applying this method to documented examples of chaotic systems. Although the method is based on harmonic linearization, a comparison of analytical results with simulation results looks promising, especially in systems with a strong low-pass filter characteristic. The main advantage, however, remains the easy application compared to complicated and exact methods. The application of this tool in diverse areas of nonlinear control theory is discussed.