We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that non-symplectic 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 or non-symplectic integrators, but in some circumstances symplecticity greatly reduces the error.

我们表明辛积分器保留了哈密顿边值问题的分支，而非辛积分器则没有。我们提供了非言语积分器打破脐带分叉的普遍描述。我们发现由某些类型的边值问题（包括经典的狄利克雷问题）引起的额外结构，对定位分支很有用。连接两点的测地线是哈密顿边值问题的一个示例，我们介绍了jet-RATTLE方法，这是一种辛积分器，可以轻松地计算测地线及其分支。最后，我们研究了周期性的干草叉分叉，这是可积哈密顿系统中产生的codimension-1分叉。辛格或非辛格积分者都没有保留它，