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 flow 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. 447--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 allow the homeomorphism $H_h$ from M into a larger manifold $M^\prime$ which contains M and is of the same dimension as M.