Preservation of bifurcations of Hamiltonian boundary value problems under discretisation

TL;DR

This paper demonstrates that symplectic integrators preserve bifurcations in Hamiltonian boundary value problems, unlike nonsymplectic methods, and introduces the jet-RATTLE method for computing geodesics and their bifurcations.

Contribution

It provides a universal framework for understanding bifurcation preservation under discretization and introduces the jet-RATTLE method for efficient bifurcation analysis.

Findings

Symplectic integrators preserve bifurcations in Hamiltonian boundary value problems.

Nonsymplectic integrators fail to preserve these bifurcations.

Symplecticity can significantly reduce errors in periodic pitchfork bifurcation computations.

Abstract

We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.