Structural Stability of Flows under Numerics

In our previous paper (SIAM J. Math. Anal.28(1997), 381–388), we showed that the qualitative properties of a Morse–Smale gradient-like flow are preserved by its discretization mapping obtained via numerical methods. In this paper, we extend the result to flows which satisfy Axiom A and the strong transversality condition. We prove that ifp 2,Φtis aCp+1 flow on a compact manifold satisfying Axiom A and the strong transversality condition, andNhis a numerical method of step sizehand orderp, then for all sufficiently smallh, there are a homeomorphismHhand a continuous real-valued functionτhonMsuch thatHh Φh+hτh(x)(x)=Nh Hh(x) andHhisO(hp)-close to the identity map onM