Dynamical analysis of evolution equations in generalized models

TL;DR

This paper formalizes the method of generalized modeling for evolution equations, providing rigorous proofs and highlighting open questions to enhance the analytical power and predictive capabilities of these models.

Contribution

It introduces a rigorous mathematical framework for generalized modeling and discusses open research questions to advance the field.

Findings

Formal mathematical foundation for generalized models

Rigorous proofs supporting key steps

Identification of open questions for future research

Abstract

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.