Detection of high codimensional bifurcations in variational PDEs
Abstract
We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as parameter-dependent semilinear Poisson equations. As an example, we consider a Bratu-type problem and show how high codimensional bifurcations such as the swallowtail bifurcation can be found numerically. In particular, our original contributions are (1) the use of the Infinite-dimensional Splitting Lemma, (2) the unified and simplified treatment of all A-series bifurcations, (3) the presentation in Banach spaces, i.e. our results apply both to the PDE and its (variational) discretization, (4) further simplifications for parameter-dependent semilinear Poisson equations (both continuous and discrete), and (5) the unified treatment of the continuous problem and its discretisation.
Cite
@article{arxiv.1903.02659,
title = {Detection of high codimensional bifurcations in variational PDEs},
author = {Lisa Maria Kreusser and Robert I McLachlan and Christian Offen},
journal= {arXiv preprint arXiv:1903.02659},
year = {2020}
}