TO BE CONTINUED ...

"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 R¹, S¹ or R², and lifts of maps on S¹, 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:

1. 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.
2. Locating fixed and periodic points for individual maps, usually using some form of Newton's method.
3. 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.
4. 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.
5. Computing basins of attraction by "escape" (or "brute force") algorithms.
6. Locating local codimension-one bifurcations, including fixed or periodic point saddle-node, period-doubling and Hopf bifurcations.
7. Continuing local bifurcations as a curve in a two-parameter familiy of maps.
8. 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.
9. Computing critical curves of a noninvertible map of the plane.
10. Computing invariant circles of maps of the plane or annulus.
11. 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.

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• Go to the Department of Mathematics and Statistics Home Page