Invariant manifolds for time-discretizations of nonlinear systems

Some computational experiments have shown that one can design difference methods for differential equations and control systems with the property that the resulting difference equations will live on, or uniformly near, a specified constraint-manifold. Here a few concrete results are given: systems that live on manifolds defined by sets of quadratic forms should be discretized by the midpoint method, which gives a system living on the same manifold; and for discretizations of autonomous Hamiltonian systems by either the midpoint or updated-Euler methods, the first few terms of an asymptotic expansion for (approximate) invariant energy-functions can be obtained easily.