Modeling with Dynamical Systems
Math 5270
Spring 2000 Syllabus
Prof. Peckham
COURSE DESCRIPTION
Solving real world problems with mathematics often involves the following
steps:
- Finding a mathematical model which reasonably approximates the
behavior of the real world problem
- Finding a mathematical solution to the (idealized) mathematical
model
- Interpreting the mathematical solution in the context of the original
problem
- Returning to step 1 if the interpretation does not appropriately fit
the data
In this course, we will concentrate on step 1, especially
for physical situations which have been found to be ``well-modeled''
by ordinary differential equations. We will also compare some of these
differential equations models to their discrete analogue,
difference equations. We will talk about step 2, the mathematical theory
of solving
differential equations, but mainly to allow us to proceed
to the interpretation in step 3. (Emphasis is placed on step 2 in Dynamical
Systems, Math 5260, next offered Fall 2001.)
Qualitative analysis (graphing solutions,
``phase planes'') and computer simulation will be emphasized over
quantitative solutions (explicit formulas).
Some specific situations we may model are:
- Population dynamics (single species, predator-prey, competing species)
- Medical dynamics (epidemics, drug ingestion/metabolism)
- Mechanical systems (pendulum, springs, celestial mechanics)
- Oscillators (biological, physical, chemical)
- Weather (Lorentz equations)
Topics are somewhat flexible and will
be adjusted according to student (and instructor) interest.