Math 5270 - Modeling with Dynamical Systems
Wednesday March 22, 2000 Midterm Topics

Spring 2000
Prof. Peckham

  1. Write differential equations with initial conditions for problems from the areas enumerated below. Indicate the meaning of all variables and parameters. Be able to indicate what assumptions lead to each term in the differential equation and how realistic those assumptions might be. Especially in cases where they might not be realistic, have an idea how adjustments to the differential equation might be made. Also have an idea of new assumptions that could be made and the resulting effect they could have on the differential equation.
    1. Radioactive decay (eg. carbon dating, ...).
    2. Population growth: Malthusian, logistic, logistic with delay, harvesting, ... .
    3. Mixing, including multiple tanks
    4. Falling bodies, including various assumptions for friction.
    5. N-body problem
    6. Chemical Reactions.
    7. Spring-mass systems, including multiple mass systems. Model a variety of assumptions for frictional forces, spring forces, external forces.
    8. Electric Circuits (RLC only).
    9. Pendulum.
    10. Interacting species: predator-prey, competition, symbiosis, ... .
    11. Epidemics: Susceptible and infected (SI, SIS), Susceptible, infected, and removed (SIR, SIRS)
  2. Write discrete population models for population growth: Malthusian, logistic, logistic with delay, harvesting, ...
  3. Analyze first order, autonomous, scalar differential equations using phase lines.
    1. Determine equilibria, linearizations near equilibria and resulting stability.
    2. Make long term predictions.
    3. Locate bifurcations in one-parameter families.
  4. Analyze first order systems of two autonomous differential equations.
    1. Determine equilibria, linearizations near equilibria and resulting stability (from the eigenvalues).
    2. Orbits corresponding to a given picture of a vector field and a given initial condition.
    3. Nullclines.
    4. Long term predictions.
    5. Properties of eigenvalues and eigenvectors of equilibrium points from phase plane pictures.
  5. Analyze first order scalar difference equations (maps) and systems of two first order difference equations. Locate fixed points and determine stability by linearization.
  6. Check to see whether a given function is a solution to a differential delay equation.
  7. Relationship between phase portraits and solution curves. Be able to construct one from the other.
  8. Convert higher order differential equations to first order systems. Convert higher order difference equations to first order systems.
  9. Any short questions related to assigned homework problems or class presentations.
  10. Other topics I think of.