MATH 5270: MODELING WITH DYNAMICAL SYSTEMS

Instructor: Professor H. Stech Telephone: 726 - 8272

Office: 156 Solon Campus Center e_mail: hstech@d.umn.edu

Office Hours: 10:00-10:50 MWF, and by appointment

Text: None. Lecture notes will be distributed. You are expected to assemble a notebook of all class handouts and assignments.

Prerequisites: An undergraduate background in calculus and elementary differential equations. Familiarity with (and access to) the Mathematica symbolic manipulation program is desirable, but Matlab, Maple, Mathcad... is also suitable.

Objectives: The effective use of "process-based" mathematical modeling requires an understanding of analytic and computer simulation methods, as well as the ability to extract from the "real world" problem under study a reliable, yet tractable, mathematical representation. Model formulation requires a familiarity with the basic physical, biological, sociological, ... principles in effect.

The goal of this course to provide an introduction to the overall modeling process by considering a variety of modeling examples taken from a wide range of disciplines. Background material and mathematical techniques for differential equations will be reviewed as each application requires.

Grading: There will be approximately 10 problem sets/coding assignments and a modeling project. There will be no formal written final exam. The modeling project will be graded according to a written paper (required of all graduate students; for extra credit for undergraduate students) and a classroom presentation. The final course grade will be determined by a weighted average of the problem set average and the modeling project grade (60% of the Problem Set Ave + 40% of the Modeling Project).

The course will be graded according to a scale no higher than

A = 90 % or above

B = 80 - 90 %

C = 70 - 80 %

D = 65 - 70 %

F = below 65 %

Points may be deducted for late assignments, or missed project deadlines.

Individuals who have and disability, either permanent or temporary, which might affect their ability to perform in this class are encouraged to inform the instructor at the start of the term. Adaptation of methods, materials, or testing may be made as required to provide equitable participation.

Tentative List of Modeling Topics

I. Introduction

1. Process-based vs empirical models

2. Model classification and characteristics

3. Why we model

II. First Order Models

1. Differential Equations Review

2. The Snow Shoveller

3. Car Deceleration

4. Population Growth Laws

III. Computer-Assisted Methods

1. Symbolic Solution Software

2. Simulation Methods with Mathematica

IV. First-Order Applications

1. Compartmental analysis

2. Lines

3. Radioactive decay (carbon dating)

4. Heating/Cooling

5. Beer’s Law and aggregation

6. Models of Motion

7. Two Snow Shovellers

8. Draining Vats (Torricelli’s Law)

9. Population Control/Harvesting

V. Simple Systems

1. Epidemic Models

2. Chemical Reactions

3. Traffic Flow

4. The Chemostat

5. Multi-Vat Systems

VI. Low-Dimensional Systems

1. Mass-Spring and Bridge Oscillations

2. Predator/Prey Systems

VII. Higher-Dimensional Systems

1. Lorenz Equations

2. Multi-Species Population Models

3. Epidemic Models

VIII. Modeling Time-Delay Systems (time permitting)

1. The Shower

2. Population Growth Models

3. Epidemic Models

4. Reaction Times