Information Technology is now a major influence on every persons life. Gradually it is being incorporated in course presentation and student learning: some lecture theatres can present computer screen images during the course of a lecture, nearly all students have access to a computer cluster.
To use computers to enhance courses suitable software must be available. It should be robust, that is, forgiving of error by the user. Software should also match the level of sophistication of the students. Many modern symbolic manipulation packages can quickly make the students loose confidence concerning their own power.
Some years ago I met Dr. Alexander Khibnik who created a very useful package TRAX . This is a user friendly program which aids the study of solution curves of ordinary differential equations or orbits of difference equations. The package requires no program writing, allows very flexible graphical output and is robust. It is possible to change parameters and see how solution behaviour changes by superimposing the new trajectories in a new colour.
These numerical experiments indicate certain parameter values are `` bifurcation '' values. Below these critical values behaviour is of one type, above the value another behaviour is dominant. For example, equilibrium points may be attractors in one range and repellors in another. Periodic solutions may be unstable and not observed in numerically generated solutions but become attracting after a bifurcation. Dr. Khibnik and a group of colleagues have released a new package, LOCBIF , to facilitate the study of these bifurcations and the numerical values of parameters which produce them.
LOCBIF follows TRAX in allowing study of solution orbits but has many additional capabilities. This package requires no program writing, the problem is entered from the keyboard using a Pascal like language with error trapping. Quite sophisticated system equations can be entered and the procedure seems robust. All functions of the package, such as graphical output formats and choice of which bifurcations are to be traced are available from pull-down menus. Although it is not a WINDOWS application the reviewer found it reasonably stable run from WINDOWS as a DOS application. Output may either be as a screen image or as a file with a summary of results. I use the WINDOWS clipboard to capture the screen image and print it from a WINDOWS application.
The package is very useful for final year undergraduate or beginning graduate courses or projects at that level. After a course has described the bifurcations commonly seen in a system with one, two, or three parameters, this package allows a study of the presence of these in parameter space. For example Hopf bifurcations can be traced together with information about the stability of the periodic solution near bifurcation. The nomenclature may be unusual since it uses Eastern European sources, Liapunov numbers etc. Luckily there is a recently published account available from Springer- Verlag .
The reviewer has found this package invaluable in convenient and rapid study of nonlinear systems. If you are conducting a course in bifurcation theory or wish to encourage demanding project work researching with non-trivial example systems which cannot be completely resolved analytically you may find it a useful aid. If you have used AUTO  this package may interest you. LOCBIF is restricted to a study of ten equations containing ten parameters but is more convenient than AUTO and gives a framework for investigating higher co-dimension bifurcations. In simple low dimension models of this type a study of most co-dimension two and some co-dimension three bifurcations is possible from a simple menu based procedure, much simpler than the iteration of edit,run,edit necessary with AUTO.
The package has a good clear manual for the study of bifurcations of equilibrium points of differential equations or fixed points of maps. A particular point to note is that there is an option for using homotopy methods to seek elusive equilibrium values. The reviewer would have liked an index, finding topics without it is time consuming and increases the frustration when you seek a topic not included. The section describing the continuation of periodic solutions and limit cycles is less well explained. A clear description of the method of obtaining the floquet multipliers in use would be useful since the choice of secant plane for a Poincare map seems critical.
The reviewer intends to purchase a multiple users site licence for this product and hopes to use it extensively with course and researchers.
 TRAX Simulation and Analysis of Dynamical Systems version 1.2.
Victor Levitin and Alexander I. Khibnik.
Distributed by Applied Biomathematics, 100 North Country Road, Setauket, New York 11733 , USA.
 LOCBIF version 2 Interactive LOCal BIFurcation Analyzer.
A. Khibnik, Y.A. Kuznetsov, V.Levitin and E.V. Nikolaev.
Distributed by CAN Expertise Centre,Kruislaan 413, 1098 SJ Amsterdam, the Netherlands.
 Dynamical Systems vol V Encyclopaedia of Mathematical Sciences: Theory of Bifurcations. Editor V.I. Arnold. Springer-Verlag, 1994.
 AUTO Software for Continuation and Bifurcation Problems. E. Doedel. 1986. California Institute of Technology.
David Knapp (D.G.Knapp@LEEDS.AC.UK).