ODE Architect: Building Order Out of Chaos

What are dynamical systems? We are surrounded by natural systems that evolve in time; many of these evolutionary processes can be modeled by differential equations. Modeling these dynamical systems with differential equations builds bridges between scientific disciplines. It not only provides a predictive tool but also a framework for examining the properties of natural systems. A wide range of questions about long-term behavior, the sensitivity of the system to data, bifurcation, chaos, and more can be addressed in the context of differential equations. Until recently, this approach was impossible in introductory courses. With the advent of affordable PC-based ODE solvers, such as the soon-to-be-released ODE Architect, however, tools are now available to interactively aid the user in building ODE models of dynamical systems and visualizing their properties.

Models, Solutions, Simulations
ODE Architect was developed, with partial support from the NSF/DUE, by the Consortium for ODE Experiments (CODEE), Intellipro, Inc., and John Wiley & Sons, Inc. CODEE saw to the mathematical side of things, Intellipro rendered CODEE's work into an interactive multimedia software package, and Wiley coordinated the efforts of both teams.

ODE Architect itself consists of three components.

The Multimedia Modeling Tool consists of 13 modules and a technical appendix. The modules span the content of the ODE course and employ animations, video, and sound to develop mathematical models and concepts in a controlled, interactive environment. Students explore the problem-solving process via what-if scenarios and explorations. A workbook to accompany the Modeling Tool has background material for each module, further explorations, and documentation on how to use the solver.

The ODE Solver Tool presents a graphical interface to enter and edit equations, control solver settings and features, and to create and edit a wide variety of graphics. Students can enter and solve their own systems of ODEs or discrete dynamical systems, input their own data tables, graph solution curves and trajectories in two or three dimensions, graph Poincaré time sections, and draw direction fields. Students can also build physical representations of systems, animate them, and save them as movies. The ODE solvers in this tool are state-of-the-art numerical solvers based on those developed by L.F. Shampine and I. Gladwell, of Southern Methodist University.

The ODE Library contains over a hundred pre-programmed ODE files covering a wide range of topics from physics, chemistry, engineering, population biology, and epidemiology. Each library file has explanatory text with the ODEs, and comes back in an active state so that the user can draw graphs of solution curves or orbits and even modify the data and the ODEs.

"No Pain, Lots of Gain"
The three components of ODE Architect described above give the user a wide range of operating modes. On a basic level, the animated multimedia modules encourage a sense of play while students are learning some important ODE concepts. In Module 2, for example, the Slope Field Golf Game challenges the user to sink a golf ball by following the flow lines generated by a slope field for a first-order ODE of the user's choice. It's great fun. (Try it!)

Beyond the basic level, the student can use ODE Architect in several ways:

Simulation
Users can enter their own systems of ODEs and use the solver to plot solution curves, orbits, component plots, etc. Alternatively, a system from a library file can be brought up and simulated 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
Sometimes 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 oscillations of linear systems with constant coefficients 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 using 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 together as a group. The students' writing skills will improve as they write up reports on their explorations. Along the way, students will also learn about the advantages and pitfalls of using numerical ODE solvers.

The Lotka-Volterra ODEs model predator and prey interactions. The graphs produced by ODE Architect (such as the one shown here) show how the predator population evolves for different sets of initial data.

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 several Wiley textbooks in calculus, differential equations, and engineering mathematics. To preview the software, or for more information, contact your local Wiley representative.