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.
- Publication:
-
Nonlinearity
- Pub Date:
- May 2020
- DOI:
- 10.1088/1361-6544/ab7293
- arXiv:
- arXiv:1903.02659
- Bibcode:
- 2020Nonli..33.2335K
- Keywords:
-
- bifurcations;
- numerical continuation;
- augmented systems;
- Bell polynomials;
- catastrophes;
- Mathematics - Numerical Analysis;
- 65P30;
- 35B32;
- 35B38;
- 35J61;
- 35J25;
- 37G10;
- 70G75;
- 34K18
- E-Print:
- Nonlinearity 33 (2020) 2335--2363