Krauskopf B., Osinga H.M.
Growing 1D and quasi 2D unstable manifolds of maps
(287K, uuencoded g-zipped postscript)
ABSTRACT. We present a new 1D algorithm for computing the global one-dimensional
unstable manifold of a saddle point of a map. This method can be
generalized to compute two-dimensional unstable manifolds of maps with
three-dimensional state spaces. Here we present a Q2D algorithm for
the special case of a quasiperiodically forced map, which allows for a
substantial simplification of the general case described in
[Krauskopf & Osinga; 1998].
The key idea is to `grow' the manifold in steps, which consist of
finding a new point on the manifold at a prescribed distance from the
last point. The speed of growth is determined only by the curvature of
the manifold, and not by the dynamics.
The performance of the 1D algorithm is demonstrated with a constructed
test example, and it is then used to compute one-dimensional manifolds
of a map modeling mixing in a stirring tank. With the Q2D algorithm we
compute two-dimensional unstable manifolds in the quasiperiodically
forced Henon map.