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 secondorder 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.

Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation

“…In [19] the authors use generating functions of the considered symplectic maps and the Lagrangian boundary conditions to assign local critical-pointsof-a-function problems to local Lagrangian boundary value problems. An introduction to generating functions can be found in [11,VI.5].…”

“…For a Lagrangian Hamiltonian boundary value problems these maps are exact, i.e. each arises as the gradient of a scalar valued map [19]. Consider a family of Hamiltonian Lagrangian boundary value problems and consider an approximation of the Hamiltoniantime-τ -map by an integrator.…”

Symplectic integration of boundary value problems

“…This applies to generic settings as well as to settings with ordinary or reversal symmetries. The conformal-symplectic symmetric case, which applies to homogeneous Hamiltonians and to the geodesic bifurcation problem in particular, is studied in [21].…”

“…are (in an appropriate equivalence relation) the only bifurcations which occur in generic Hamiltonian boundary value problems with up to three parameters [20].…”

“…In [20] the authors reveal how the completely integrable structure and the structure of the boundary conditions induce a Z/2Z-symmetry in the generating function of the problem family. The singular point of a pitchfork bifurcation is unfolded under the presence of a Z/2Z-symmetry to a pitchfork bifurcation.…”