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arXiv · 1903.02659

Detection of high codimensional bifurcations in variational PDEs

Lisa Maria Kreusser·Robert I McLachlan·Christian Offen
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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.

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Lisa Maria Kreusser, Robert I McLachlan, Christian Offen. 2019-03-06. Detection of high codimensional bifurcations in variational PDEs. https://doi.org/10.1088/1361-6544/ab7293

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