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.

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.

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.

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 / sec^{2}. 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).

- The length of the stick:
*l*. - The length of the bar connecting the moving flywheel to the base
of the inverted pendulum:
*d*. - The radius of the flywheel:
*r* - The frequency of the uniform circular motion of
the flywheel:
*f*.

- Derive the differential equation satisfied by the angle formed by the inverted pendulum with the vertical direction
- Write an Octave or Matlab scripts which allow you to integrate the equations of motion.
- Write an Octave/MATLAB
*function*named is_stable which for given parameters (l, d, r, f) will return 0 or 1, depending on whether the motion of the inverted pendulum is stabilized in the nearly-vertical position or not.

- hill.m - the specification of the differential equation.
- hill_unstable.m - an M-file that calls
*ode45*.

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

- test.m gives sample usage of several functions for testing values of vectors
- stick.m and xypos.m show how to generate one frame of the animation of the inverted pendulum shown in class
- expression_formats_session.txt shows a Maxima session used to convert expressions to useful formats.

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.

The zip archive hw4.zip should be submitted by clicking here or using WebDAV.

Marek Rychlik <rychlik@u.arizona.edu> Last modified: Wed Aug 27 19:41:25 MST 2003