Structural stability of Morse-Smale gradient-like flows under discretizations

In this paper, we show that the qualitative property of a Morse-Smale gradient-like flow is preserved by its discretization mapping obtained via numerical methods. This means that for all sufficiently small h, there is a homeomorphism H-h conjugating the time-h map Phi(h) of the how to the discretization mapping phi(h). Garay [Numer. Math., 72 (1996), pp. 449-479] showed this result by relying on techniques of Robbin [Ann. Math., 94 (1971), pp. 441-493]. Our result sharpens and unifies that in [Numer. Math., 72 (1996), pp, 449-479] by using Robinson's method in [J. Differential Equations, 22 (1976), pp. 28-73] of the structural stability theorem for diffeomorphisms. We also study the problem on a manifold with boundary. Under the assumption that the manifold M is positively invariant for the flow, we show that the qualitative properties are weakly stable, which means we anew the homeomorphism H-h from M into a larger manifold M' which contains M and is of the same dimension as M.