"To Be Continued..." (TBC) is a dynamical systems investigation
tool, written primarily by Dr. Bruce Peckham with significant help from several
undergraduate and graduate research assistants. For a brief history and
list of contributors, see
History of the Software and its Authorship.

As the name suggests, the software
is an ongoing project. It is written in C++, using
OpenGL for 2D and 3D graphics, and XFORMS for its graphical user interface
(GUI). There is currently a version which runs on Linux machines and a version
which runs on Silicon Graphics machines (IRIX).

*TBC* is currently capable of handling dynamical systems
defined by maps on **R**^{1}, **S**^{1} or **R**^{2},
and lifts of maps on **S**^{1}, all with an arbitrary number
of parameters. It has the advantage (and disadvantage) that it was developed
for the study of maps rather than differential equations. It was also developed
to concentrate on families of maps of the plane -- either two-parameter
families or families with more parameters but with only two designated
as primary parameters.

*TBC* is capable of the following:

- Iteration
- Forward iteration of points or sets via invertible or noninvertible maps.
- Backward iteration of points or sets via invertible maps.
- Backward iteration of maps with multiple inverses by specifing a single branch of the inverse.
- Cobweb iteration of 1D maps.
- Locating fixed and periodic points for individual maps, usually using some form of Newton's method.
- Continuing fixed or periodic points to curves (corresponding to one-parameter families or one-parameter cuts through a larger parameter space) and surfaces (corresponding to two-parameter families or two-parameter slices of parameter space). Certain global surfaces can be computed in a single run by choosing appropriate parametrizations of the surfaces.
- Computing one-dimensional stable and unstable manifolds of saddle fixed or periodic points. A fundamental interval of the unstable eigenspace can be iterated to obtain the global unstable manifold. For invertible maps, a fundamental interval of the stable eigenspace can be iterated backward to obtain the global stable manifold.
- Computing basins of attraction by "escape" (or "brute force") algorithms.
- Locating local codimension-one bifurcations, including fixed or periodic point saddle-node, period-doubling and Hopf bifurcations.
- Continuing local bifurcations as a curve in a two-parameter familiy of maps.
- Locating homoclinic and heteroclinic points. As with the fixed and periodic points, individual points, curves and surfaces of homoclinic and heteroclinic points can be computed.
- Computing critical curves of a noninvertible map of the plane.
- Computing invariant circles of maps of the plane or annulus.
- Computing codimension-one curves and codimension-two points for noninvertible bifurcations of maps of the plane. Codimension-one curves: "cusp transition'' and ``loop creation on an unstable manifold.'' Codimension-two point: ``cusp-cusp'' point. See `Unfolding the cusp-cusp bifurcation of planar endomorphisms," with Bernd Krauskopf and Hinke Osinga, SIAM Journal on Dynamical Systems, Vol. 6 (2), 2007, pp403-440 for more information.

Users may define their own maps by modifying a single file (maps.c) and recompiling.

With some C programming familiarity, users may add new bifurcations to locate and continue. With more expertise, users may write additional modules to attatch to the existing software. (Instructions for these last two user modifications are not yet complete. 7/17/02)

Acknowledgement: Some of the development of this software was supported by the National Science Foundation through grants DMS-9020220, DMS-9505051, and DMS-9973926.

This page is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Thursday, 23-Oct-2008 22:51:46 CDT.