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Detection of high codimensional bifurcations in variational PDEs. (English) Zbl 1487.37097

Summary: 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.

MSC:

37M20 Computational methods for bifurcation problems in dynamical systems
65P30 Numerical bifurcation problems
47J15 Abstract bifurcation theory involving nonlinear operators
35B32 Bifurcations in context of PDEs
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