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Hinke's interest lies in the computation of invariant manifolds for
dynamical systems. Most of this work is in collaboration with
Bernd
Krauskopf, University of Bristol. The main idea behind the
algorithms is described in the paper Twodimensional global unstable manifolds
of maps (published electronically
and in Int. J. Bifurcation & Chaos 8(3):
483503, 1998).


Algorithms for maps 

Two special algorithms are derived from the paper Twodimensional global unstable manifolds of maps. Software for onedimensional global unstable manifolds of maps is available for use inside the DsTool environment; see the IMApreprint and the paper for the IMA proceedings.  
The ideas for onedimensional and twodimensional manifolds are
combined in a specialized algorithm for twodimensional manifolds in
quasiperiodically forced systems. The paper Growing 1D and
quasi 2D unstable manifolds of maps describes both the
onedimensional and the quasi twodimensional technique (published
in J. Comput. Phys. 146(1): 404419, 1998); see also the
animations. The quasi 2D algorithm
is used in the study of boundary crisis in
quasiperiodically forced systems, research that is done together
with Ulrike
Feudel.


Algorithms for vector fields 


The algorithm is also specialized for vector fields. The boundary of the computed part of the manifold is propagated further in steps that amount to solving a boundary value problem. Both unbounded manifolds and manifolds that converge to an attractor can be computed. Examples and animations are the twodimensional stable manifold of the Lorenz system and the twodimensional unstable manifold of Arneodo's system. This work will be published in Twodimensional global manifolds of vector fields, CHAOS 9(3), 1999, with a multimedia supplement deposited in EPAPS. 
The above two examples are both threedimensional vector fields. The computation of twodimensional manifolds can straightforwardly be generalized to vector fields of higher dimension. The idea of growing stable and unstable manifolds using the formulation of a boundary value problem is generalizable to higher dimensional manifolds as well. However, the implementation and visualization of such a high dimensional object is quite complicated. More details are given in Global manifolds of vector fields: The general case.  
The same algorithm can be used for the computation of stable and unstable manifolds of periodic orbits. In this case, the manifold can be nonorientable and be topologically equivalent to a Möbius strip. How to adapt the algorithm to nonorientable manifolds is described in Nonorientable manifolds of periodic orbits. 