STRUCTURAL STABILITY OF FLOWS UNDER NUMERICS

In our previous paper (SIAM J. Math. Anal. 28 (1997), 381-388), we sho wed that the qualitative properties of a Morse-Smale gradient-like flo w are preserved by its discretization mapping obtained via numerical m ethods. In this paper, we extend the result to flows which satisfy Axi om A and the strong transversality condition. We prove that if p great er than or equal to 2, Phi(t) is a C-p + 1 flow on a compact manifold satisfying Axiom A and the strong transversality condition, and N-h is a numerical method of step size h and order p, then for all sufficien tly small h, there are a homeomorphism H-h and a continuous real-value d function tau(h) on M such that H-h circle Phi(h+h tau h(x))(x) = N-h circle H-h(x) and H-h is O(h(p))-close to the identity map on M. (C) 1997 Academic Press.