Home Page for TBC:
TO BE CONTINUED ...
A Discrete Dynamical Systems Software Package
Main Author: Bruce B. Peckham
"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:
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
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.
TBC provides several precoded maps for investigation. These include:
the Henon map (several versions), the quadratic map (real and complex versions:
z ->z2+C), Arnold's Standard Circle Map Family: x -> x + b + a sin(x),
the delayed logistic map,
forced oscillator caricature maps, ... .
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.
Users may define their own maps by modifying a single file (maps.c) and
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
This page is maintained by Bruce Peckham (firstname.lastname@example.org)
and was last modified on Thursday, 23-Oct-2008 22:51:46 CDT.