NUMERICAL CHAOS, SYMPLECTIC INTEGRATORS, AND EXPONENTIALLY SMALL SPLITTING DISTANCES

Two discrete versions of Duffing's equation are derived as reductions of discretizations of the nonlinear Schrödinger (NLS) equation. Inheriting the properties of the NSL discretizations, one is shown to be integrable and, using a Mel'nikov type analysis, the other discretization is shown to be nonintegrable. Arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation, as h → 0. This suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous Hamiltonian systems. © 1993 Academic Press, Inc.