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Discussion: Chaos lesson
Topic: Chaos in the Classroom
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Subject:   RE: Chaos in the Classroom
Author: Sione
Date: Feb 26 2004
Greetings to all,

I have watched this topic "Chaos in the classroom" for a while
and wondered whether to respond or not. One reason for deciding not
to respond is that I am not a teacher. Since I do not know of
what is taught (math topic) and at what level for high school, I
hesitated to respond just in case I made a wrong comment about the
US high school math curriculum. I am outside of the US , so I do not
know much about the US high school math curriculum at all.

I would like to make some comments:

- "Chaos" is a discipline of Computational Physics (mathematical
physics). The main branch is called "Non-Linear Dynamics" (NLD) in
which "Chaos" is a subclass of NLD. There is another subclass of NLD
called "Bifurcation" where it is similar to "Chaos" but have different
conditions for their stationary points and attractors.

- The foundation of "Chaos" or NLD in general is the study of the theory
of non-linear differential equations.

- I believe that this subject is too complex to teach at high school
level , and in my opinion, if it is not part of the math curriculum,
then do not teach them or even discuss it in classroom.
WHY? The theory of NLD is very complex to explain to high school kids.
Chaos is not random events, it is deterministic in its nature but
unpredictable, because of its non-periodic flow. Probability Theory(PT)
involves non-deterministics concept, so chaos and NLD are different from

- I do not know what Non Linear Dynamic (NLD) model that the applet
implements but I suspect that it is "Lotka-Volterra" population growth.
My point in not recommending the teaching of the subject at high school
is that when kids look at those applets they immediately think : RANDOM
EVENTS,PROBABILITY, CHANCE etc. NLD is non of those. If then the kids
would ask that if NLD (Chaos) is not random events then what is it . Say
the teacher tells them that they are "deterministic model but
unpredictable" . Well , this statement will confuse even adults. I
believe that this subject should be left to where it is usually taught.
It is taught at university level for math courses in differential
calculus, stability problems in physics, and also in engineering
science. At high school, telling kids about non-linear differential
equations (that is what CHAOS is) will be far too complex at that level.

- I have developed a software module to be part of a larger finance
application for predicting (forecasting) the stock market price options.
This was a work for a local finance company. The model I used was the
"GLASS-LACKEY" chaotic time-series. Stock market price option is chaotic
(unpredictable) over time. I did develop a neuron-fuzzy (Neural Network &
Fuzzy Logic Hybrids) module to tune the "CLASS-LACKEY" differential
equations for a better precision of forecast. Runge-Kutta of order 4 was
the numerical methods that was developed to solve the solutions for the
"CLASS-MACKEY" differential equations.

- Non Linear Dynamics is not a specialist area for me, but often I
develop numerical codes (software) if it requires for a particular task.
There are some members of our group (Scientific Computing) here at
Auckland University, New Zealand who are specialist in Non Linear
Dynamics. One researcher is applying NLD and chaos to designing
"industrial control systems" , another member is researching the
stability of lasers in telecommunications networks (fiber optics
communication systems) and this same person is developing an embedded
systems software (signal processing) to predict the wave patterns from a
coma patient at intensive care in hospital. This helps doctors to tell
when to administer drugs and what dose, by reading the brain wave
pattern. Brain waves from coma patient at hospital is chaotic
(unpredictable). One is doing R&D in NLD application in Computational

- Chaos and Non Linear Dynamics will be growing in industrial
applications over the coming years. This is a ground breaking discipline that
will be adopted by growing number of industrial technologists now and the

The following is a link to an article(PDF document) in Non-Linear-Dynamics
(NLD) from one of the leading expert in the field of NLD.

   "Exploring Complex Networks" by Steven Stogatz


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