ODE Architect: Performance and Versatility
ODE Architect is a suite of software tools published by John Wiley & Sons that
can be used to supplement learning and enhance understanding in a sophomore
ordinary differential equations course. This tool is one of many that have been
made possible because of more powerful and widely available computers, but more
importantly because of the growing infrastructure of mathematicians able to
utilize their full capabilities.
The classical obstacles to students in trying to understand differential equations
is the difficulty in achieving an intuition for what the solution should be
from looking at the equation. Most of us, having learned the subject in the
traditional way, have a limited toolkit of examples that we truly understand.
Unlike Maple and Mathematica, products that solve differential equations analytically
or numerically, ODE Architect is a specific tool that focuses on qualitative
aspects of numerical solutions.
Three distinct features highlight this package, the Discrete Tool for finding
fixed points and fractals, the ODE Architect Tool for visualizing solutions
to one-, two-, and three-dimensional systems, and the Model Builder Tool for
building, animating, and analyzing physical representations of dynamical systems.
The Discrete Tool performs fixed-point iterations, producing spider or ladder
diagrams. For example, the user inputs relation x=f(x), and some initial value
x0. Discrete Tool computes the iterates xn+1= f(xn) and graphs them in the xy-plane
together with the function y=f(x) and the line y=x. While this tool is not as
sophisticated as the other two, it is the only tool of its kind available for
this type of plot.
Additionally, Discrete Tool can be used to compute dimensional fixed-point
plots and will also generate fractals with complex iterations.
The second tool, the ODE Architect Tool, is a remarkable piece of software
to help students beginning to learn the geometry of ODE solutions. The main
screen is divided into three sections. The first is for the user input of the
differential system itself. It may contain parameters, which must also be numerically
defined. The second section is a control panel where initial conditions are
entered and the system is solved. The third section displays the solution. For
example for the system x.=cy and y.=d x, with both c and d defined and initial
conditions entered, one of the multitude of displays available is the solution
graphs in the tx- and ty-planes. Alternatively, the solution graph in the xy-plane
is available.
A second highly useful function within the ODE Architect Tool is the Sweep
tool. With respect to the system above, a family of solutions of this system
can be displayed on a single graph for the values of c in some user input range.
For example, we could specify a starting c = 1 and stopping c = 2 and #Points
= 11. The result is the 11 solutions corresponding the c-values 1.0, 1.1, 1.2,
…, 2.
The result is the family of solutions in the ty-plane. A similar family of
solutions is in the tx-plane. It is also possible to animate the solution, that
is, to watch the solution traced out over time (t).
An especially well-conceived
feature is that solutions compound on the graph; the viewer sees all solutions
to date, even if the differential equation is changed. Of course, from time
to time it becomes necessary to Clear All Runs and begin anew. Many other user
controls such as line width and color are available to enhance the output view.
Plots, which copy and paste in vector mode, can be scaled as needed. They can
also be annotated.
The avid Maple user may claim that Maple can do this just as well, and in some
cases better, than the Sweep tool. While it is true that an experienced Maple
user could write a package, say MapleArchitect, to do this for the student,
ODE Architect is a specialized tool with an intuitive interface that puts control
in the hands of the student–an important advantage.
The third tool in ODE Architect suite is the Model Builder, and the features
of this tool are definitely not possible within the Maple engine. In brief,
the Model Builder allows users to link properties such as location and rotation
of graphics objects with the dynamic variables in a system of ordinary differential
equations. The Model Builder helps users make deeper connections between a system
of differential equations and the physical model that it represents. While there
is a tutorial to get the user started, this tool is not easy to use.
In summary, while ODE Architect is overall simple to use, some of the parameters
are difficult to set. There is a definite learning curve, but one that most
students can navigate. The program contains an extensive library of existing
systems, single ODEs and models ready-made and easily modified, and an accompanying
report writer (actually WordPad) is built in. Here students can copy and annotate
plots they have made, fill in text, and create reports. Students and faculty
giving ODE Architect a fair chance will be impressed with its performance and
versatility.
