Topology of first integrals via Milnor fibrations II

TL;DR

This paper extends the application of Milnor's Fibration Theory from isolated to non-isolated singularities in dynamical systems, offering geometric-topological classifications of first integrals using advanced Singularity Theory tools.

Contribution

It introduces a novel approach to classify foliations with first integrals, including non-isolated singularities, by connecting harmonic morphisms with Milnor fibrations.

Findings

Topological descriptions of foliations with first integrals.

Analysis of solution graphs for quasilinear systems.

Extension of classification methods to non-isolated singularities.

Abstract

This survey is the continuation of a series of works aimed at applying tools from Singularity Theory to Differential Equations. More precisely, we utilize the powerfull Milnor's Fibration Theory to give geometric-topological classifications of first integrals of differential systems. In the previous paper, systems of first-order quasilinear partial differential equations were examined, focusing on the case of an isolated singularity. Now, we address both cases of isolated and \textit{non-isolated singularities} for more general dynamical systems (namely, \textit{foliations}) that admit at least one first integral. For this, we utilize recently established connections between harmonic morphisms and Milnor fibrations to provide topological and geometric descriptions of the foliations under consideration. In particular, we apply these results to analyze the graph of solutions of some quasilinear systems.