Structural stability on basins for numerical methods

Abstract

In this paper, we show that a flow W with a hyperbolic compact attracting set is structurally stable on the basin of attraction with respect to numerical methods. The result is a generalized version of earlier results by Garay, Li, Pugh, and Shub. The proof relies heavily on the usual invariant manifold theory elaborated by Hirsch, Pugh, and Shub (1977), and by Robinson (1976). 1. STATEMENT OF THEOREMS Let M be a smooth complete Riemannian manifold with a distance d arising from the Riemannian metric and Diff(M) be the set of diffeomorphisms on M with the strong topology and distance d0l. A flow is a map Ap: IR x M -M that satisfies the group property: ps(pt(x)) = fs+t(x) A set A is attracting for a flow (p on M if there is a neighborhood U of A such that pT(Cl(U)) C int(U) for some T > 0 and A = n (pt(U). The basin of A is the t>o set B (A) = {x E M: lim d((pt(x), A) = 0}, where d(y, A) = min{d(y, z): z E A}. t-.oo An attracting set for a flow is closed and invariant (see [18]). A compact invariant set A for a flow (p on M is hyperbolic if the restriction of the tangent bundle TM of M to A splits into three continuous subbundles, TMIA = EU (D Es (D Span(X), invariant under the derivative of opt, Dp t, such that Dfpt expands EU and Dcpt contracts ES. Here X is the vector field induced by the flow P. Definition 1. For p > 1, let p be a CP+1 flow on M. A CP+1 function N: R x M -M is called a numerical method of order p for p if there are positive constants K and ho such that d(ph(x), Nh(x)) 4 (see [1]). Received by the editors January 28, 1997 and, in revised form, May 6, 1997. 1991 Mathematics Subject Classification. Primary 58F10, 58F12, 65L20, 34D30, 34D45.