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.
For example, the implicit midpoint rule achieves this for the critical energy surface of the H´enon-
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.
McLachlan, R.I., Perlmutter, M., Quispel, G.R.W. (2001), On the nonlinear stability of symplectic integrators, Research Letters in the Information and Mathematical Sciences, 2, 93-107