# Detection of high codimensional bifurcations in variational PDEs

## TL;DR

This paper develops a numerical method to detect high codimensional bifurcations in parameter-dependent PDEs, using bifurcation test equations and the Infinite-dimensional Splitting Lemma, applicable to both continuous and discretized problems.

## Contribution

It introduces a unified approach for detecting A-series bifurcations in variational PDEs, simplifying analysis and computation in Banach spaces.

## Key findings

- Successfully detects swallowtail bifurcations in a Bratu-type problem.
- Provides a unified framework applicable to PDEs and their discretizations.
- Simplifies the analysis 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.