cs249 lecture notes, Fall 2001
Week 9, Monday

No class on Thursday.  Get together to work on the
assignment if you like.


Chaos
-----

A dynamical law is one that predicts the future behavior
of a system based on the current or initial conditions.

Classic example is Newton's three laws of motion.

Success of Newtonian mechanics led to the Principle of
Determinism, which is that all physical systems are

   deterministic: given the current state of the system,
   we can predict the future state, for any length of time.

   More specifically, the Principle of Determinism assumes
   that the accuracy of the answer depends on the accuracy
   of the initial conditions.

   Even more specifically, it was commonly assumed that
   doubling the accuracy of the initial estimates would
   double the accuracy of the predictions.

This turns out to be FALSE (at least some of the time).

For many systems, the error in the prediction increases
quickly (exponentially) with time.

1) short-term predictions can be accurate

2) long-term predictions are no better than chance

3) better initial conditions might give the ability
   to predict SLIGHTLY longer into the future, but
   they do not improve the accuracy of the prediction
   after that point.

These systems are said to be "dynamically unstable" or
CHAOTIC.

	Warning: this is a specific, technical definition
	of "chaotic".  It does NOT mean disorderly or
	apparently random.  Some chaotic systems are
	apparently orderly, and they are all deterministic,
	but they cannot be predicted accurately.


Why not just measure initial conditions perfectly?

1) as an experimental matter, that's impossible

2) as a modeling matter, it's irrelevant

   even if the measurement were exact, there is always
   SOMETHING in the model that you left out.

   Why did you leave it out?  Because you assumed that
   it would have a small effect on the outcome.
   
   In chaotic systems, even the smallest perturbation can
   have a wild effect on the result:  "Butterfly effect"

3) In numerical computations, we are constantly perturbing
   the system with roundoff errors and truncation errors
   (as in Euler's method).


Chaos is a trendy topic in mathematics, and has been
written up in several popular treatments, including
most notably

Chaos : Making a New Science, by James Gleick


Predator-prey model
-------------------

What does all this have to do with us?

Predator-prey model is an example of a non-linear dynamical
system.

It is a particularly interesting one because it is
sometimes chaotic and sometimes not, depending on the
parameter r.

For some values of r, the population converges on an stable
    fixed point.

For some values, it converges on a stable periodic fixed point.

If the system converges to a stable fixed point, is it chaotic?


For some values of r the system is chaotic, which means that
future values are very sensitive to initial conditions.


A quick MATLAB hint
-------------------

To draw a column of points at location r:

    // red is a vector of population values
    // r is the parameter that generated red

    rvec = r * ones (1, length(red));
    h = plot (rvec, red, 'r.');
    set (h, 'MarkerSize', 3);