---
_id: '19935'
abstract:
- lang: eng
  text: 'A bifurcation is a qualitative change in a family of solutions to an equation
    produced by varying parameters. In contrast to the local bifurcations of dynamical
    systems that are often related to a change in the number or stability of equilibria,
    bifurcations of boundary value problems are global in nature and may not be related
    to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed
    framework which studies the bifurcations of critical points of functions. In this
    paper we study the bifurcations of solutions of boundary-value problems for symplectic
    maps, using the language of (finite-dimensional) singularity theory. We associate
    certain such problems with a geometric picture involving the intersection of Lagrangian
    submanifolds, and hence with the critical points of a suitable generating function.
    Within this framework, we then study the effect of three special cases: (i) some
    common boundary conditions, such as Dirichlet boundary conditions for second-order
    systems, restrict the possible types of bifurcations (for example, in generic
    planar systems only the A-series beginning with folds and cusps can occur); (ii)
    integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic
    pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing
    symmetries can exhibit restricted bifurcations associated with the symmetry. This
    approach offers an alternative to the analysis of critical points in function
    spaces, typically used in the study of bifurcation of variational problems, and
    opens the way to the detection of more exotic bifurcations than the simple folds
    and cusps that are often found in examples. '
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. Bifurcation of solutions to Hamiltonian boundary value
    problems. <i>Nonlinearity</i>. 2018:2895-2927. doi:<a href="https://doi.org/10.1088/1361-6544/aab630">10.1088/1361-6544/aab630</a>
  apa: McLachlan, R. I., &#38; Offen, C. (2018). Bifurcation of solutions to Hamiltonian
    boundary value problems. <i>Nonlinearity</i>, 2895–2927. <a href="https://doi.org/10.1088/1361-6544/aab630">https://doi.org/10.1088/1361-6544/aab630</a>
  bibtex: '@article{McLachlan_Offen_2018, title={Bifurcation of solutions to Hamiltonian
    boundary value problems}, DOI={<a href="https://doi.org/10.1088/1361-6544/aab630">10.1088/1361-6544/aab630</a>},
    journal={Nonlinearity}, author={McLachlan, Robert I and Offen, Christian}, year={2018},
    pages={2895–2927} }'
  chicago: McLachlan, Robert I, and Christian Offen. “Bifurcation of Solutions to
    Hamiltonian Boundary Value Problems.” <i>Nonlinearity</i>, 2018, 2895–2927. <a
    href="https://doi.org/10.1088/1361-6544/aab630">https://doi.org/10.1088/1361-6544/aab630</a>.
  ieee: R. I. McLachlan and C. Offen, “Bifurcation of solutions to Hamiltonian boundary
    value problems,” <i>Nonlinearity</i>, pp. 2895–2927, 2018.
  mla: McLachlan, Robert I., and Christian Offen. “Bifurcation of Solutions to Hamiltonian
    Boundary Value Problems.” <i>Nonlinearity</i>, 2018, pp. 2895–927, doi:<a href="https://doi.org/10.1088/1361-6544/aab630">10.1088/1361-6544/aab630</a>.
  short: R.I. McLachlan, C. Offen, Nonlinearity (2018) 2895–2927.
date_created: 2020-10-06T16:28:36Z
date_updated: 2022-01-06T06:54:14Z
department:
- _id: '636'
doi: 10.1088/1361-6544/aab630
extern: '1'
language:
- iso: eng
main_file_link:
- url: https://doi.org/10.1088/1361-6544/aab630
page: 2895-2927
publication: Nonlinearity
publication_identifier:
  issn:
  - 0951-7715
  - 1361-6544
publication_status: published
status: public
title: Bifurcation of solutions to Hamiltonian boundary value problems
type: journal_article
user_id: '85279'
year: '2018'
...