---
_id: '19938'
abstract:
- lang: eng
text: '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. '
article_type: original
author:
- first_name: Robert I
full_name: McLachlan, Robert I
last_name: McLachlan
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: https://orcid.org/0000-0002-5940-8057
citation:
ama: 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
apa: 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
bibtex: '@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} }'
chicago: '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.'
ieee: 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.
mla: 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.
short: R.I. McLachlan, C. Offen, Foundations of Computational Mathematics 20 (2020)
1363–1400.
date_created: 2020-10-06T16:31:46Z
date_updated: 2022-01-06T06:54:14Z
department:
- _id: '636'
doi: 10.1007/s10208-020-09454-z
extern: '1'
intvolume: ' 20'
issue: '6'
language:
- iso: eng
main_file_link:
- url: https://rdcu.be/b79aB
page: 1363-1400
publication: Foundations of Computational Mathematics
publication_status: published
status: public
title: Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation
type: journal_article
user_id: '85279'
volume: 20
year: '2020'
...