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:
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.
- 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.
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
- 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
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.
What to turn in?
The zip archive hw4.zip should be submitted by clicking here or using WebDAV.
Marek Rychlik <email@example.com>
Last modified: Wed Aug 27 19:41:25 MST 2003