Differential Equations

• Errors in ODE integration schemes as a function of method and step size. Empirical measures of the order of a method.
• Experiment with graphical integrators and then argue for the correctness of some of the results seen. (e.g. fences, vertical asymptotes, behavior at infinity.)
• Bifurcations of 2d homogeneous linear systems.
• Matrix exponential and applications.
• Computer assistance for variation of parameters.
• Computer assistance for change of coordinates.
• Logistic, predator-prey and generalizations as a function of parameter.
• Computer algebra systems in scratchpad mode to assist with power series and Frobenius methods solutions.
• Use computers to bring some special functions to life.
• Study some basic low codimension bifurcations (e.g. saddle node, pitchfork, hopf.)
• Sensitive dependence on initial conditions.
• Study attractive periodic points as a function of parameter in the 1d discrete logistic equation to see chaotic behavior.
• Explore the restricted 3 body problem numerically.
• Web Resources:

  Java and Matlab versions of Dfield and Pplane:

  http://math.rice.edu/$\tilde{\ }$polking

  Dynamics Solver Source:

  http://tp.lc.ehu.es/JMA/ds/ds.html

  Interactive Differential Equations:

  http://awlonline.com/ide/ (All labs available online in pdf format!)

  IDEA:

  http://www.sci.wsu.edu/idea

  CODEE

  http://www.math.hmc.edu/codee/main.html

  Mathlab:

  http://www.mathlab.cornell.edu/computer_and_portfolio/index.html#ODE