On the nonlinear stability of symplectic integrators

The modified Hamiltonian is used to study the nonlinear stability of symplectic integrators, especially for nonlinear oscillators. We give conditions under which an initial condition on a compact energy surface will remain bounded for exponentially long times for sufficiently small time steps. While this is easy to achieve for non-critical energy surfaces, in some cases it can also be achieved for critical energy surfaces (those containing critical points of the Hamiltonian). For example, the implicit midpoint rule achieves this for the critical energy surface of the Henon-Heiles system, while the leapfrog method does not. We construct explicit methods which are nonlinearly stable for all simple mechanical systems for exponentially long times. We also address questions of topological stability, finding conditions under which the original and modified energy surfaces are topologically equivalent.