# Modeling and Visualization with ODE Architect: An Example Set

ODE Architect at Wiley & Sons is a software tool to enhance the learning of dynamical systems, both continuous and discrete. It was created by Intellipro and is currently being published by Wiley, who bundles it with the ODE textbooks by Boyce & diPrima and Borrelli & Coleman. It runs on Windows PCs only. (ODE Toolkit is a related cross-platform project written in Java.)

The examples in this collection illustrate some of the features of the ODE Architect. These units are independent of one another and for convenience are divided into three groups: elementary, intermediate, and advanced. Each unit includes several screen shots from ODE Architect. There are many other accessible examples and models within the ODE Architect package itself. The ODE Library in the tool contains over one hundred briefly described, illustrated, and editable examples appropriate for differential equations courses.

This material is available as individual chapters in Portable Document Format (PDF).

### Elementary Units

Lascaux Cave Paintings
Radioactive decay. Using computer model to approximate age. Backward IVP. Scaling.
Vertical Motion
Uses Newton's Second Law to model vertical motion. Scaling. ODE Architect is used to answer the question "Does a body take longer to rise or fall?". Solution formula is not helpful in answering this question if air resistance is taken into account.
Sprint
Models strategies for running 100 meters. Compares results with real data.
Samurai
Uses a computer model to design an efficient cutting instrument. Also derives a solution formula in polar coordinates which describes the sword's edge mathematically.

### Intermediate Units

Linear system modeled with the Balance Law. Examines sensitivity of solutions to the environmental parameters. ODE Architect will find eigenvalues and steady-states.
Coaxial Cable
Uses on-off functions as inputs to a simple RC-circuit for sending a coded message. Examines sensitivity of output to the frequency of the input.
Air-Conditioning
Uses Newton's Law of Cooling to describe how an air-conditioner keeps a room's temperature within a prescribed range.
Sky Diver
Determines when a sky diver should pull the rip cord to minimize descent time (and survive). ODE Architect solves the problem by overlapping graphs of a forward IVP and a backward IVP.
Things may go wrong even when using a good numerical ODE solver: aliasing, extension, choice of step-size, etc.

Van der Pol
A nonlinear model with a limit cycle. Effect of parameter on shape and period of the limit cycle, and animation of Hopf bifurcation. ODE Architect automatically calculates the eigenvalues of the linearized system.
Lorenz
ODE Architect finds equilibrium points and eigenvalues of the linearized Lorenz system and graphically illustrates how parameter changes affect stability. 3-D graphs of chaotic wandering, period doubling sequences.
Battle of the Bulge
Actual combat data from the battle is used to determine the coefficients for a combat model. The modeling linear system is solved by ODE Architect and the results are compared with the actual data.
Autocatalator
Interacting chemical species in an autocatalytic reaction lead to a nonlinear system which exhibits peculiar oscillatory behavior. ODE Architect shows how changing the rate constants can be used to turn the oscillations on and off.
Satiable Predation
Predator-prey model where the predator's appetite satiates. A sensitivity study shows a Hopf bifurcation followed by a reverse Hopf bifurcation. Illustrated via animation.
Rotational Stability
Examines the stability of steady rotations of a rigid body (like a book or tennis racket) when tossed in the air. 3-D graphs clearly shows which steady rotations are stable and which are unstable.
Fitzhugh-Nagumo (written by Michael E. Moody)
Uses ODE Architect to show how a planar system of ODEs representing a neuron responds to a stimulus.