The cusp map in the complex-frequency plane for absolute instabilities

It is well known that absolute instabilities can be located by prescribed mappings from the complex-frequency plane to the wavenumber plane through the dispersion relation D(ω,k)=0. However, in many systems of physical interest the dispersion relation is polynomial in ω while transcendental in k, and the implementation of this mapping procedure is particularly difficult. If one maps consecutive deformations of the Fourier integral path (originally along the real k axis) into the ω plane, points having (∂D/∂k)=0 are readily detected by the distinctive feature of their local maps. It is shown that a simple topological relationship between these points and the image of the real k axis determines the stability characteristics of the system, without mapping from the ω plane back into the k-plane.