Title: Bifurcation of solutions to Hamiltonian boundary value problems,
Abstract: 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.