Math 481/581, Assignment 4 (Draft 5)

Deadline: April 15, 2005


This assignment and assignment 3 together constitute Project 2.


To integrate systems of first order differential equations using a built-in ODE solver of Octave or MATLAB. To plot time series and phase diagrams and draw conclusions from these plots concerning equilibrium states, periodic solutions and chaotic behavior.


Predator-prey population model

In class we investigated the pair of coupled logistic equations. The relevant files are:

Investigate the question whether the populations always go to an equilibrium, or perhaps whether they can oscillate. You are allowed to change the constants in the model.

Forced pendulum with friction

In class we investicated the forced pendulum with friction, where the forcing is periodic:

It was suggested in class that by weakenning the damping term and choosing the forcing term suitable, and possibly changing the length of the pendulum one can achieve chaotic motion.

For other values of the parameters you may observe simple periodic motion following a transient phase during which the pendulum settles on its asymptotic behavior.

Stabilizing inverted pendulum

As you know, it is possible to balance a long pole on the top of your head. You can also try to balance a pencil on a tip of your finger. If you ever tried, you know that the shorter the object, the harder the task. Does stabilizing a pole or a pencil require "intelligence"? In other words, is it necessary to react to the falling pole by some, perhaps complex, feedback mechanism?

Let us consider the following model:

This figure was created with XFig. You can obtain the source here: inverted_pendulum.fig.

Assume gravity constant g=9.8 meters / sec2. Also, assume that the mass is m=1 kg. Thus, the remaining parameters are specified using the metric system (lengths in meters, time in seconds, frequency in Hertz=1/sec).

For simplicity, assume that there is no damping.

MATLAB commands for this assignment

If you are working with MATLAB, you will not find the command lsode. I provided two files which illustrate the use of the command ode45 in MATLAB which illustrate the use of ode45:

Maxima calculation of the second derivative

The file script.maxima provides the code which computes the second derivative of the position of the base

More useful tricks

A paper

Write a paper (say, 5-7 pages) discussing your findings. Try to find similar research on the Web.

Your paper should include the plots of interesting solutions of the models. Make sure to include all files you used in the solution in your submission. Also, include all relevant references to external documents (textbooks, papers, Web pages) in BibTeX format.

What to turn in?

The zip archive should be submitted by clicking here or using WebDAV.

Marek Rychlik <>
Last modified: Wed Aug 27 19:41:25 MST 2003