Generalized models provide a framework for the study of evolution equations without specifying all functional forms. The generalized formulation of problems has been shown to facilitate the analytical investigation of local dynamics and has been used successfully to answer applied questions. Yet their potential to facilitate analytical computations has not been realized in the mathematical literature. In the present paper we introduce the method of generalized modeling in mathematical terms, supporting the key steps of the procedure by rigorous proofs. Further, we point out open questions that are in the scope of present mathematical research and, if answered could greatly increase the predictive power of generalized models.
Figure 1: Bifurcation diagram in generalized parameter space. The red surface indicates Hopf bifurcation and the blue surface
saddle-node bifurcations. Higher co-dimension bifurcations are indicated by separate labels. The main codimension two curves are Gavrilov-Guckenheimer (GG), Takens-Bogdanov (TB) and double-Hopf (DH) bifurcations.