Math 5270 - Modeling with Dynamical Systems
Friday April 28, 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 (-> midterm 2)
    6. Chemical Reactions. (-> midterm 2)
    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). (-> midterm 2 -> next course)
    9. Pendulum.
    10. Interacting species: predator-prey, competition, symbiosis, ... .
    11. Epidemics: Susceptible and infected (SI, SIS), Susceptible, infected, and removed (SIR, SIRS) (-> midterm 2)
  2. Write discrete population models for population growth: Malthusian, logistic, logistic with delay, harvesting, ...
  3. 1D DE's: first order, autonomous, scalar differential equations.
    1. Determine equilibria, linearizations near equilibria and resulting stability.
    2. Make long term predictions (phase lines).
    3. Locate bifurcations in one-parameter families. Identify saddle-node, transcritical and pitchfork bifurcations.
  4. 2D DE's: first order systems of two autonomous differential equations.
    1. Determine equilibria, linearizations near equilibria and resulting stability (from the eigenvalues).
    2. Make long term predictions (phase plane).
    3. Nullclines.
    4. Properties of eigenvalues and eigenvectors of equilibrium points from phase plane pictures.
    5. Understand what stable and unstable manifolds of a saddle fixed point are. Recognize where they are in a phase portrait.
    6. Understand what a Hopf bifurcation is.
    7. Understand what a homoclinic bifurcation is.
    8. Understand the Determinant-Trace space and its relationship to the stability of an equilibrium point.
  5. 1D maps: first order scalar difference equations.
    1. Locate fixed points and determine stability by linearization.
    2. Determine fixed points and their stability from the graph of the recursion relation.
    3. Cobweb iteration.
  6. 2D Maps: systems of two first order difference equations.
    1. Locate fixed points and determine stability by linearization.
    2. Understand the Determinant-Trace space for 2D maps and its relationship to the stability of a fixed point.
    3. Understand what a homoclinic tangle is and why it is important (existence of chaos).
  7. Delay differential equations
    1. Check to see whether a given function is a solution to a differential delay equation.
    2. Locate any equilibrium solutions
    3. Linearize around an equilibrium solution
    4. Determine the stability of an equilibrium solution if the linearization turns out to be either a 1D (linear) differential equation without delay, or if it turns out to be of the form dx/dt = a x(t-r).
  8. Relationship between phase portraits and solution curves. Be able to construct one from the other.
  9. Convert higher order differential equations to first order systems. Convert higher order difference equations to first order systems.
  10. Interpret solutions to DE's and Maps in the context of a model.
  11. Any short questions related to assigned homework problems or class presentations.
  12. Other topics I think of.