For a diffeomorphism F on R-2, it is possible to find periodic orbits of F
of period k by applying Newton's method to the function F-k - I, where I is
the identity function. (We actually use variants of Newton's method which
are more robust than the traditional Newton's method.) For an initial point
x, we iterate Newton's method many times. If the process converges to a po
int p which is a periodic point of F, we say x is in the Newton basin of p
for period k, denoted by B(p, k). We investigate the size of the Newton bas
in and how it depends on p and k. In order to understand the basins of high
period orbits, we choose p a periodic point of F with period k, then we in
vestigate basins B(p, nk) for n = 1, 2, 3,.... We show that if p is an attr
acting orbit, then there is an open neighborhood of p that is in all the Ne
wton basins B(p, nk) for all n. If p is a repelling periodic point of F, it
is possible that p is the only point which is in all of the Newton basins
B(p, nk) for all n. It is when p is a periodic saddle point of F that the N
ewton basin has its most interesting behavior. Our numerical data indicate
that the area of the basin of a periodic saddle point p is proportional to
lambda(c) where lambda is the magnitude of the unstable eigenvalue of DFk (
p) and c is approximately -1 (c approximate to -0.84 in Fig. 5). For long p
eriods (k more than about 20), many orbits of F have lambda so large that t
he basins are numerically undetectable. Our main result states that if p is
a saddle point of F, the intersection of Newton basins B(p, nk) of p inclu
des a segment of the local stable manifold of p. (C)2000 Elsevier Science B
.V. All rights reserved.