Example Set for ODE Architect

This scanned document is broken up into smaller sections for convenience.

• Front Matter
• Elementary Units

1. Lascaux Cave Paintings: Radioactive decay. Using computer model to approximate age. Backward IVP. Scaling.
2. 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.
3. Sprint: Models strategies for running 100 meters. Compares results with real data.
4. Samurai Sword: 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

1. Lead in the Body: Linear system modeled with the Balance Law. Examines sensitivity of solutions to the environmental parameters. ODE Architect will find eigenvalues and steady-states.
2. 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.
3. Air-Conditioning: Uses Newton's Law of Cooling to describe how an air-conditioner keeps a room's temperature within a prescribed range.
4. 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.
5. Good Solver, Bad Behavior: Things may go wrong even when using a good numerical ODE solver: aliasing, extension, choice of step-size, etc.

• Advanced Units
  

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. Fitzhugh-Nagumo: Uses ODE Architect to show how a planar system of ODEs representing a neuron responds to a stimulus.