Qualitative property between flows and numerical methods Original Research Article

In this paper, we show that qualitative properties of Axiom A flows with the strong transversality condition are preserved by its discretization mapping obtained via numerical methods. More precisely, if ϕtϕt is an Axiom A flow with the strong transversality condition and NN is a numerical method of order p⩾1p⩾1 for ϕtϕt on a smooth compact manifold MM, then there exists T>1T>1 such that for all sufficiently large n∈Nn∈N, there are a continuous onto function Hn:M→MHn:M→M and a continuous function τn:M→Rτn:M→R such that ϕτn(x)∘Hn(x)=Hn∘(NT/n)n(x)ϕτn(x)∘Hn(x)=Hn∘(NT/n)n(x) for all x∈Mx∈M.