Finding all periodic orbits of maps using Newton methods: sizes of basins

For a diffeomorphism F on R2, it is possible to find periodic orbits of F of period k by applying Newton’s method to the function Fk−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 point 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 basin 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 investigate basins B(p,nk) for n=1,2,3,… . We show that if p is an attracting orbit, then there is an open neighborhood of p that is in all the Newton 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 Newton 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 λc where λ is the magnitude of the unstable eigenvalue of DFk(p) and c is approximately −1 (c≈−0.84 in Fig. 5). For long periods (k more than about 20), many orbits of F have λ so large that the 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 includes a segment of the local stable manifold of p.