Bifurcation of solutions to Hamiltonian boundary value problems

TL;DR

This paper investigates global bifurcations of solutions in Hamiltonian boundary value problems using singularity theory, revealing how boundary conditions and symmetries influence bifurcation types and introducing new bifurcation phenomena.

Contribution

It applies geometric and singularity theory methods to analyze bifurcations in Hamiltonian boundary value problems, highlighting effects of boundary conditions and symmetries on bifurcation types.

Findings

Boundary conditions restrict bifurcation types to folds and cusps.

Integrable systems can exhibit a novel periodic pitchfork bifurcation.

Symmetries lead to restricted bifurcations related to the system's symmetry.

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.