Structural Stability for the Euler Method

Source abstract

In this paper, we show that if a flow $\varphi^t$ has a hyperbolic chain recurrent set either without fixed points or with only fixed points, and satisfies the strong transversality condition, then $\varphi^t$ is structurally stable with respect to numerical methods, including the Euler method, which was not done in [B. M. Garay, Numer. Math., 72 (1996), pp. 449--479], [M.-C. Li, J. Differential Equations, 141 (1997), pp. 1--12], and [M.-C. Li, SIAM J. Math. Anal., 28 (1997), pp. 381--388]. The proof is an application of the invariant manifold techniques developed by Hirsch, Pugh, and Shub [M. Hirsh, C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, New York, 1977] and Robinson [C. Robinson, J. Differential Equations, 22 (1976), pp. 28--73]. The result is an extension of our previous work [M.-C. Li, Proc. Amer. Math. Soc., 127 (1999), pp. 289--295].