Singularities are ubiquitous in dynamical systems. They often mark boundaries between different dynamical regimes and also serve as organizing centers for the geometry of phase space and parameter space. In this project, we aim to extend geometric desingularization methods developed in the context of continuous-time systems to various classes of discrete-time maps.
The project aims to study geometric desingularization methods of non-hyperbolic fixed points for various classes of iterated maps with a focus in multiple time scale problems. Iterated maps are discrete-time dynamical systems and non-hyperbolic fixed points occur if the linearization of the dynamics does not locally dominate higher-order nonlinear terms. For continuous-time dynamical systems, one available method to analyze nonhyperbolic equilibria is geometric desingularization, i.e., blowing up an equilibrium to a co-dimension one manifold usually taken as a sphere. We are going to investigate several classes of iterated maps, where developing a suitable blow-up in the discrete-time context is expected to be a highly effective tool in dynamical systems. The key class of motivating problems are multiscale maps with different time scales.