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Christian Offen

Doctor of Philosophy, (Mathematics)
Study Completed: 2020
College of Sciences

Citation

Thesis Title
Analysis of Hamiltonian boundary value problems and symplectic integration

Mathematical models using parameter-dependent differential equations can be used to describe different natural processes. For many models, at certain parameter values, called bifurcation points, the model behaviour changes dramatically (planets changing their course, for example, or price levels jumping spontaneously). Mr Offen developed a geometrical framework which relates bifurcations in certain situations (known as Hamiltonian boundary value problems) to classical catastrophe theory. He derived a classification of bifurcation phenomena, discovered novel bifurcation phenomena, and demonstrated implications for numerical computations to make model predictions more reliable.

Supervisors
Distinguished Professor Robert McLachlan
Dr David Simpson

Publications

Published papers

(6) (with Kreusser, L., McLachlan, R.) Detection of high codimensional bifurcations in variational PDEs, Nonlinearity (2020), Volume 33, Number 5, 2335–2363, https://doi.org/10.1088/1361-6544/ab7293

(5) (with McLachlan, R.) Preservation of bifurcations of Hamiltonian boundary value problems under discretisation, Foundations of Computational Mathematics (2020), https://doi.org/10.1007/s10208-020-09454-z

(4) (with McLachlan, R., Tapely, B.) Symplectic integration of PDEs using Clebsch variables, Journal of Computational Dynamics (2019), Volume 6, Number 1, 111-130, https://dx.doi.org/10.3934/jcd.2019005

(3) (with McLachlan, R.) Symplectic integration of boundary value problems, Numerical Algorithms (2019), Volume 81, 1219–1233, https://doi.org/10.1007/s11075-018-0599-7

(2) (with McLachlan, R.) Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci, Volume 48 (2018), 83-99, New Zealand Journal of Mathematics, https://arxiv.org/abs/1804.07479

(1) (with McLachlan, R.) Bifurcation of solutions to Hamiltonian boundary value problems, Nonlinearity, Volume 31, Number 6 (2018), https://doi.org/10.1088/1361-6544/aab630

Submitted papers

(8) (with McLachlan, R.) Backward error analysis for variational discretisations of partial differential equations, https://arxiv.org/abs/2006.14172

(7) Local intersections of Lagrangian manifolds correspond to catastrophe theory, https://arxiv.org/abs/1811.10165