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									                                                                                                   FOCUS        ON CALCULUS                 5

ODE Architect:                                                                                     5
                                                                                                                Predator population x(t)

Building Order Out of Chaos

Robert L. Borrelli and Courtney S. Coleman, Harvey Mudd College


D     ynamical systems are becoming an
      important component of many
                                                and more can be addressed in the context
                                                of differential equations. Until recently,

courses in the lower division curriculum,       this approach was impossible in introduc-          2
especially the introductory ordinary dif-       tory courses. With the advent of afford-
ferential equations (ODEs) course. The          able PC-based ODE solvers, such as the
reason for this is that dynamical systems       soon-to-be-released ODE Architect, how-            1
are the glue that holds the current interdis-   ever, tools are now available to interac-
ciplinary trend together.                       tively aid the user in building ODE mod-
   What are dynamical systems? We are           els of dynamical systems and visualizing           0
surrounded by natural systems that evolve       their properties.                                      0    5         10       15      20   25
in time; many of these evolutionary pro-                                                                                   t
cesses can be modeled by differential equa-     Models, Solutions, Simulations                 Figure 1 The Lotka-Volterra ODEs model
tions. Modeling these dynamical systems            ODE Architect was developed, with           predator and prey interactions. The graphs
with differential equations builds bridges      partial support from the NSF/DUE, by           produced by ODE Architect (such as the one
between scientific disciplines. It not only     the Consortium for ODE Experiments             shown here) show how the predator population
provides a predictive tool but also a frame-    (CODEE), Intellipro, Inc., and John Wiley      evolves for different sets of initial data.
work for examining the properties of natu-      & Sons, Inc. CODEE saw to the math-
ral systems. A wide range of questions          ematical side of things, Intellipro rendered
                                                                                               based on those developed by L.F.
about long-term behavior, the sensitivity       CODEE’s work into an interactive multi-
                                                                                               Shampine and I. Gladwell, of Southern
of the system to data, bifurcation, chaos,      media software package, and Wiley coor-
                                                                                               Methodist University.
                                                dinated the efforts of both teams.
                                                   ODE Architect itself consists of three         The ODE Library contains over a hun-
                                                components.                                    dred pre-programmed ODE files covering
 About CODEE                                                                                   a wide range of topics from physics, chem-
   The Consortium for Ordinary                     The Multimedia Modeling Tool con-
                                                                                               istry, engineering, population biology, and
                                                sists of 13 modules and a technical appen-
  Differential Equations Experiments                                                           epidemiology. Each library file has ex-
                                                dix. The modules span the content of the
 (CODEE) distributes information on                                                            planatory text with the ODEs, and comes
                                                ODE course and employ animations,
 the design and use of interactive                                                             back in an active state so that the user can
                                                video, and sound to develop mathemati-
 computer experiments in courses                                                               draw graphs of solution curves or orbits
                                                cal models and concepts in a controlled,
 involving ODEs. CODEE runs faculty                                                            and even modify the data and the ODEs.
                                                interactive environment. Students explore
 workshops, publishes a newsletter,             the problem-solving process via what-if        “No Pain, Lots of Gain”
 and is now completing work on ODE              scenarios and explorations. A workbook            The three components of ODE Architect
 Architect. The sponsors and faculty            to accompany the Modeling Tool has back-       described above give the user a wide range
 members associated with CODEE are:             ground material for each module, further       of operating modes. On a basic level, the
   Cornell University: John Hubbard             explorations, and documentation on how         animated multimedia modules encourage
 and Beverly West; Harvey Mudd                  to use the solver.                             a sense of play while students are learning
 College: Robert Borrelli (PD/PI) and              The ODE Solver Tool presents a graphi-      some important ODE concepts. In Module
 Courtney Coleman (Co-PI), Michael              cal interface to enter and edit equations,     2, for example, the Slope Field Golf Game
 Moody; Rensselaer Polytechnic                  control solver settings and features, and to   challenges the user to sink a golf ball by
                                                create and edit a wide variety of graphics.    following the flow lines generated by a
 Institute: William Boyce and William
                                                Students can enter and solve their own         slope field for a first-order ODE of the
 Siegmann; St. Olaf College: Arnold
                                                systems of ODEs or discrete dynamical          user’s choice. It’s great fun. (Try it!)
 Ostebee and Matthew Richey; Stetson                                                              Beyond the basic level, the student can
                                                systems, input their own data tables, graph
 University: Michael Branton and                solution curves and trajectories in two or     use ODE Architect in several ways:
 Margie Hale; Washington State                  three dimensions, graph Poincaré time             Simulation Users can enter their own
 University: Michael Kallaher, Thomas           sections, and draw direction fields. Stu-      systems of ODEs and use the solver to plot
 LoFaro, and Kevin Cooper; West                 dents can also build physical representa-      solution curves, orbits, component plots,
 Valley College: Douglas Campbell and           tions of systems, animate them, and save       etc. Alternatively, a system from a library
 Wade Ellis.                                    them as movies. The ODE solvers in this        file can be brought up and simulated
                                                tool are state-of-the-art numerical solvers                             continued on page 6

ODE Architect
continued from page 5

using the suggestions in its banner. A
favorite of ours is the animated Hopf
bifurcation which is displayed in the
Satiable Predator library file.
   Discoveries and Conjectures Some-
times a simulation reveals features of a
dynamical system that would be hard to
see in any other way. An example is the
comparison of rise and fall times of a ball
under air resistance in Module 5. (Just try
to do the math!)
   Illustration of Theory Forced oscilla-
tions of linear systems with constant coef-
ficients come up a lot in the applications.
Comparing the input and output of such a
system on the screen is very instructive.
   Modeling ODE Architect makes it easy
to see the effect on the output of a system
when various modeling assumptions are
used. For example, in Module 5 the effects
of viscous and Newtonian damping on a
falling body are compared.
   There are also intangible benefits to us-
ing a versatile software package such as
ODE Architect. It is natural for students to
work in teams while exploring ODEs that
model physical situations. This enhances
communication as they learn to work to-
gether as a group. The students’ writing
skills will improve as they write up re-
ports on their explorations. Along the way,
students will also learn about the advan-
tages and pitfalls of using numerical ODE

Multiple Course Uses
   ODE Architect is designed for use with
any ODE text, introducing and enhancing
the modeling and visualization approach.
It can be used to support the ODE portion
of a calculus course, as part of a regular
ODE course at the sophomore level, a
supplement to an engineering systems
course, or as part of advanced ODE or
dynamical systems courses. As expected,
ODE Architect coordinates well with sev-
eral Wiley textbooks in calculus, differen-
tial equations, and engineering mathe–
   To preview the software, or for more
information, contact your local Wiley
representative or send e-mail to
math@wiley.com. L

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