Symplectic integration of boundary value problems

TL;DR

This paper investigates the use of symplectic integrators for Hamiltonian boundary value problems, showing they preserve bifurcations while nonsymplectic methods do not, highlighting their importance beyond long-term dynamics.

Contribution

It demonstrates that symplectic integrators preserve bifurcations in Hamiltonian boundary value problems, a property not shared by nonsymplectic integrators, extending their applicability.

Findings

Symplectic integrators preserve bifurcations in Hamiltonian boundary value problems.

Nonsymplectic integrators do not preserve these bifurcations.

Symplecticity is relevant even for short-time boundary value problems.

Abstract

Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to {\em long-time} behaviour. They are directly connected to the dynamical behaviour of symplectic maps $\varphi\colon M\to M$ on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map $\varphi\colon M\to M'$ which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.