Singularities in bivariate normal mixtures

We investigate mappings $F=(f_1,f_2)\text{\hspace{0.17em}⁣}:{\mathbb{R}}^2\to {\mathbb{R}}^2$ where $f_1,f_2$ are bivariate normal densities from the perspective of singularity theory of mappings, motivated by the need to understand properties of two-component bivariate normal mixtures. We show a classification of mappings $F=(f_1,f_2)$ via $\mathcal{A}$-equivalence and characterize them using statistical notions. Our analysis reveals three distinct types, each with specific geometric properties. Furthermore, we determine the upper bounds for the number of modes in the mixture for each type.