Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation

R.I. McLachlan, C. Offen, Foundations of Computational Mathematics 20 (2020) 1363–1400.

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Journal Article | Published | English
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Abstract
We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.
Publishing Year
Journal Title
Foundations of Computational Mathematics
Volume
20
Issue
6
Page
1363-1400
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McLachlan RI, Offen C. Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation. Foundations of Computational Mathematics. 2020;20(6):1363-1400. doi:10.1007/s10208-020-09454-z
McLachlan, R. I., & Offen, C. (2020). Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation. Foundations of Computational Mathematics, 20(6), 1363–1400. https://doi.org/10.1007/s10208-020-09454-z
@article{McLachlan_Offen_2020, title={Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation}, volume={20}, DOI={10.1007/s10208-020-09454-z}, number={6}, journal={Foundations of Computational Mathematics}, author={McLachlan, Robert I and Offen, Christian}, year={2020}, pages={1363–1400} }
McLachlan, Robert I, and Christian Offen. “Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation.” Foundations of Computational Mathematics 20, no. 6 (2020): 1363–1400. https://doi.org/10.1007/s10208-020-09454-z.
R. I. McLachlan and C. Offen, “Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation,” Foundations of Computational Mathematics, vol. 20, no. 6, pp. 1363–1400, 2020.
McLachlan, Robert I., and Christian Offen. “Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation.” Foundations of Computational Mathematics, vol. 20, no. 6, 2020, pp. 1363–400, doi:10.1007/s10208-020-09454-z.

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