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Module/Chapter 8: Compartment Models
Assumed concepts: Systems of ODEs
Both the module and the chapter use 1D, 2D, 3D, and 4D applications (in sequence) to illustrate principles of the Balance Law and interpretations of solutions. The final submodule introduces Hopf bifurcations and the interesting behavior of chemical reactions in an autocatalator. Three of the models are linear; the last is nonlinear.
Module/Chapter 9: Population Models
Assumed concepts: Systems of ODEs
The module and chapter introduce simple 1D, 2D, and 3D nonlinear models, and give a discussion of the biology behind the models.
Module/Chapter 10: The Pendulum and Its Friends
Assumed concepts: Systems of ODEs; the first submodule of Module 4; the arctangent function; parametric curves on a surface
The pendulum submodule explores all the traditional aspects of a pendulum, using integrals of motion. Child on a Swing and Geodesics on a Torus give new extensions of pendulum analysis; supporting detail is given in the chapter. The approach to modeling is a little different in this chapterófor example, how to invent functions that behave as needed (Child on a Swing), or how to exploit part of an ODE that looks familiar (Geodesics on a Torus).
Module/Chapter 11: Applications of Series Solutions
Assumed concepts: Systems of ODEs; acquaintance with infinite series and convergence; the first submodule of Module 4
The module introduces the techniques and limitations of series solutions of second-order linear ODEs. The Robot and Egg provides motivation for the subject and Aging Springs illustrates Bessel functions. The chapter contains information about the mathematics of series solutions.
Module/Chapter 12: Chaos and Control
Assumed concepts: The pendulum ODEs ofModule 10; systems ofODEs; experience with Poincare sections and/or discrete dynamical systems (Chapter 13) is helpful
The three submodules of this unit tell a story, and in the process illustrate a theorem from current research. This module uses sensitivity to initial conditions and the Poincare section to assist with the analysis. Sinks, saddles, basins, and stability are described. Finally, the elusive boundaries of the Tangled Basin provide a mechanism for control of the chaotically wandering pendulum. The module ends in a fascinating control game that is both fun to play and illuminates the theorem mentioned above.
Module/Chapter 13: Discrete Dynamical Systems
Assumed concepts: Acquaintance with complex numbers and the ideas of equilibrium and stability are helpful
The module provides a gentle introduction to an increasingly important subject. The chapter fills in the technical and mathematical background.
This module could be used successfully in a liberal arts course for students with no calculus.
Level-of-Difficulty of Modules
The chart below is a handy reference for the levels of the submodules.
Elementary Intermediate Advanced
2.1, 2.2 2.3, 2.4
3.1 3.3 3.2
5.1 5.2 5.3
6.1 6.2 6.3
7.1 7.2 7.3
8.1 8.2 8.3 8.4
9.1 9.2 9.3
10.1 10.2 10.3
12.1 12.2, 12.3
13.1 13.2 13.3
In constructing this chart we have used the following criteria: Elementary: Straightforward, self-contained, can be used as a unit in any introductory calculus or ODE course.
Intermediate: Builds on some prior experience, including earlier submodules and chapters.
Advanced: More challenging models or mathematics, especially suitable for term or group projects.
The Userís Guide is the basic reference for the features of the ODE Architect Tool. The Guide is included on the CD-ROM for ODE Architect.
1 Modeling with the ODE Architect
Douglas Campbell & Wade Ellis, West Valley Community College
Building a Model of the Pacific Sardine Population 2 The Logistic Equation 10 Introducing Harvesting via Landing Data 12 How to Model in Eight Steps 15 Explorations 17
Margie Hale & Michael Branton, Stetson University
Differential Equations 26
Solutions to Differential Equations 26
Solving a Differential Equation 27
Slope Fields 27
Initial Values 28
Finding a Solution Formula 28
The Juggler 30
The Sky Diver 31
Margie Hale, Stetson University & Douglas Quinney, University of Keele
Newton's Law of Cooling 44
Cooling an Egg 44
Finding a General Solution 44