Geometric Desingularization of Higher Codimension Singularities in Fast-Slow Systems

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Multiple time scale dynamical systems occur in a wide variety of applications and are a cornerstone in the mathematical analysis of singularly perturbed differential equations. Here we propose to study the theoretical aspects of two-time scale dynamics, so-called fast-slow systems, focusing on higher-codimension singularities. The work proposed here has two main mathematical motivations: on the one hand there is an abstract classification problem of fast-slow systems according to their singularities, which originates from singularity theory and Takens' work on constrained differential equations; on the other hand there are new and recent discoveries on adaptive networks that hint towards the great relevance of singularities beyond the classical fold, cusp, transcritical, etc\'etera. More precisely, we are going to develop parts of local unfoldings for fast-slow systems with singular points for the fast subsystem, arising from two different scenarios: $D_4^\pm$-singularities (umbilics), and a four-dimensional fast-slow Bogdanov-Takens singularity. Furthermore, we propose to apply the knowledge gained from the previous studies to two models of fast-slow adaptive networks: the first is a fast-slow consensus motif with three dynamic weights, while the second is concerned with a rivalry network. In such models, the singularities involved need at least two slow variables for the full analysis of the dynamics, i.e., they are of higher codimension in comparison to the classical folded singularities. We are going to use geometric desingularization via blow-up as our main tool to understand trajectories and invariant manifolds near each singularity. This will be combined with dynamical systems techniques such as stability theory, center manifolds, asymptotic analysis of special equations in scaling charts, bifurcation theory, and variational equations. The outcomes of the project are going to be substantially extended unfolding results for fast-slow singularities, new reduction techniques for higher-dimensional fast-slow systems, and a deeper geometric understanding of the role and properties of singularities in adaptive networks with two time scales.