---
res:
bibo_abstract:
- "Ordinary differential equations (ODEs) and partial differential equations (PDEs)
arise\r\nin most scientific disciplines that make use of mathematical techniques.
As exact solutions are in general not computable, numerical methods are used to
obtain approximate\r\nsolutions. In order to draw valid conclusions from numerical
computations, it is crucial\r\nto understand which qualitative aspects numerical
solutions have in common with the\r\nexact solution. Symplecticity is a subtle
notion that is related to a rich family of geometric properties of Hamiltonian
systems. While the effects of preserving symplecticity\r\nunder discretisation
on long-term behaviour of motions is classically well known, in this\r\nthesis\r\n(a)
the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian\r\nboundary
value problems is explained. In parameter dependent systems at a bifurcation\r\npoint
the solution set to a boundary value problem changes qualitatively. Bifurcation\r\nproblems
are systematically translated into the framework of classical catastrophe theory.
It is proved that existing classification results in catastrophe theory apply
to\r\npersistent bifurcations of Hamiltonian boundary value problems. Further
results for\r\nsymmetric settings are derived.\r\n(b) It is proved that to preserve
generic bifurcations under discretisation it is necessary and sufficient to preserve
the symplectic structure of the problem.\r\n(c) The catastrophe theory framework
for Hamiltonian ODEs is extended to PDEs\r\nwith variational structure. Recognition
equations for A-series singularities for functionals on Banach spaces are derived
and used in a numerical example to locate high-codimensional bifurcations.\r\n(d)
The potential of symplectic integration for infinite-dimensional Lie-Poisson systems
(Burgersâ€™ equation, KdV, fluid equations, . . . ) using Clebsch variables is analysed.\r\nIt
is shown that the advantages of symplectic integration can outweigh the disadvantages
of integrating over a larger phase space introduced by a Clebsch representation.\r\n(e)
Finally, the preservation of variational structure of symmetric solutions in multisymplectic
PDEs by multisymplectic integrators on the example of (phase-rotating)\r\ntravelling
waves in the nonlinear wave equation is discussed.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Christian
foaf_name: Offen, Christian
foaf_surname: Offen
foaf_workInfoHomepage: http://www.librecat.org/personId=85279
orcid: https://orcid.org/0000-0002-5940-8057
dct_date: 2020^xs_gYear
dct_language: eng
dct_publisher: Massey University@
dct_title: Analysis of Hamiltonian boundary value problems and symplectic integration@
...