A catalogue of singularities
Abstract
This paper is an attempt to classify finite-time singularities of PDEs. Most of the problems considered describe free-surface flows, which are easily observed experimentally. We consider problems where the singularity occurs at a point, and where typical scales of the solution shrink to zero as the singularity is approached. Upon a similarity transformation, exact self-similar behaviour is mapped to the fixed point of a {\it infinite dimensional dynamical system} representing the original dynamics. We show that the dynamics close to the fixed point is a useful way classifying the structure of the singularity. Specifically, we consider various types of stable and unstable fixed points, centre-manifold dynamics, limit cycles, and chaotic dynamics.
Cite
@article{arxiv.0711.0442,
title = {A catalogue of singularities},
author = {Jens Eggers and Marco A. Fontelos},
journal= {arXiv preprint arXiv:0711.0442},
year = {2007}
}