Global Manifolds 1D is software that can be used with the software package DsTool, developed at the Center for Applied Mathematics, Cornell University. It is written for the Tcl/Tk version of DsTool, which can be downloaded from http://www.math.cornell.edu/~gucken/software.html. (See also the entry on DsTool
DsTool allows to compute fixed points of a map and their one-dimensional stable and unstable manifolds by computing the orbits of several along the eigendirection, close to the fixed point.
It would be nicer to have the points ordered (and hence plotted) according to the arclength distance to the fixed point, i.e. similar to the manifold for a vector field. The Global Manifolds 1D code grows the manifolds by arclength distance to the fixed point, and distributes the points according to the local curvature of the manifolds. Furthermore, it offers the possibility to compute only one side of the manifold.
This new upgraded version of the Global Manifolds 1D software package is capable of computing one-dimensional stable manifolds of two-dimensional discrete systems without requiring explicit or approximate knowledge of the inverse map. The computations are done using forward iterates only, so numerical inaccuracies due to approximating the Jacobian matrix or using Newton's method are avoided.
|References to Papers|
J. England, B. Krauskopf and H.M. Osinga (2003)
Computing one-dimensional stable manifolds of planar maps without the inverse
Bristol Centre for Applied Nonlinear Mathematics preprint 2003.02.
B. Krauskopf and H.M. Osinga (2000)
Investigating torus bifurcations in the forced Van der Pol oscillator
in E.J. Doedel, L.S. Tuckerman (Eds.) Numerical Methods for Bifurcation
Problems and Large-Scale Dynamical Systems, IMA Volumes in Mathematics
and its Applications 119, pp. 199-208, Springer-Verlag.
B. Krauskopf and H.M. Osinga (1998)
Growing 1D and quasi 2D unstable manifolds of maps (abstract)
J. Comput. Phys. 146(1): 404-419.