Toward a solution to a problem of Poincaré: A Macro-analysis of geometric variation of high-dimensional dynamics

Around the turn of the 20th century, Poincaré formulated the idea of studying nature via the qualitative, geometric study of spaces of mappings we use to model nature. Since then, much of mathematical dynamics as well as nonlinear dynamics in many applied fields has worked to achieve partial solution to this problem. In this talk I will begin by discussing a construction that provides a means of attacking Poincaré's original problem. In a practical way, this goal will be achieved using a function space (neural networks) that admits a measure. Using the chosen function space, a Monte Carlo analysis relative to this measure of the macroscopic geometric features will be presented. In particular, the geometric quantification will consist of analyzing a function that measures the number of positive Lyapunov exponents (and hence expanding directions) with parameter variation. This function is then rescaled to remove a dependence on dimension and the number of parameters such that an analysis can be performed in the asymptotic limit of a large number of dimensions.