[sdiy] slightly ot: Chua circuit and chaos

[Czech Martin]

First: I'm not the author. It's some Mr. Wehr.

The author claims that research in very nonlinear dynamics,
colloquial called "chaos theory" has some serious
methodical and other problems. I can not write about it all,
but only some thoughts (again, I'm not clever enough to
come up with these ideas, it's the work of Mr. Wehr,
but perhaps I'm clever enough to think about it
and discuss it here).

1.: Real Numbers:
Is "chaos" (only) a logical consequence of the properties
of real, or to be more specific irrational or
transcendental numbers? 
Example: 1/6= 0.166666666666...
         PI=3.14159265...
         SQRT(2)=1.41421356...
This example shows, that rational numbers have a simple, 
predictable decimal representation. PI and SQRT(2) have not.
It can be shown that both numbers can not be expressed
as ratio of two integers. The sport of computing 
such numbers to ever more valid digits shows, that
there is no order in the sequence of digits, it's
really "chaos", or unpredictable. You can not have an algorithm
that can show all digits at once.
Now, a number of chaotic mathematical sequences are developed
my some modulo function. I.e. some digits are thrown away,
others appear. From said property of the non rational numbers
it is clear that such sequences will be "chaotic" if the
starting value is non rational.
Furthermore: the non rational numbers are dense, it is very likely 
that you hit one as starting value.
Still today the infinite properties of the non rational numbers
are a puzzle for mathematicians. Or as Konecker is believed to say:
"Integers were made by God, al else is devils work".

We simply use these numbers for every day engineering.
Are we allowed to do so?
Is the concept of real numbers a good model for what's
really going on?

Striking example is "Logistic Equation", or foxes and rabbits.
These or other equations are used to describe the mutual
dependencies and development of predator-prey relationships.
Mr. Wehr states that if integers were used for the description
instead of real numbers (which make some sense, because
you can have 1 rabbit, but not 0.845 rabbits), all
chaotic effects disappear. Of course, the integer description
needs other forms of equations, too.


2. Differential Equations (D.E.): Peano's theorem states that all
D.E. that are interesting for us here
have one, and only one solution. This solution is unique.
Or in terms of state trajectories: trajectories to not
intersect and are a sharp line of flight, not fuzzy.
This is the deeper reason why systems smaller then 3rd
order can not be chaotic: on a 2D plane there is not
enough "room" for the trajectories not to intersect.

The ubiquitous use of D.E. is sometimes misleading.
D.E. can be chaotic, but is a D.E. a good model for
a given system? This is not always true, as fox/rabbit
example shows.

3. "Chaos Theory": "Chaos Theory" as such doesn't exist. 
Much work has been done, some considerable insight
but so far there is no solid body of theory that -say-
90% of the experts would agree as substantial.
So far there seems to be no strict mathematical definition of 
what chaos is and what it is not.
If this is true, how can we say that the experimental results
e.g. of Chua's circuit are "chaotic"?
We should not use computer simulation to "prove" that.

4. Hype: "Chaos" was a great hype in the 80s-90s. Suddenly everything
was chaotic, a new "world concept" was promised.
People from other fields of science or even people who
had no clue about math at all wrote papers and books
that can not withstand even 10 seconds of careful though.
In fact nonlinear dynamics is not so new. Lagrange is often
badly cited: "give me the world D.E. and the current state,
and I will predict the future". This is often cited to show
that the head of the "deterministic paradigm" did not
know about nonlinear difficulties in prediction. Mr.
Wehr seems to show that the opposite is true.
Lagrange was fully aware of the complications of nonlinear
D.E. This became even more important when Poincare proved
the unpredictability of the trajectories of planets in 
solar systems with more then one planet.

5. Computers: Because the use of computer D.E. solvers
gives  sometimes totally wrong and misleading results the
feedback of experiment is very important.
It is important to validate if the physics/mathematical transformation
is valid, if your solving of the math problem is valid and 
if your reverse transformation math/physics is valid.
It is all important. 
Many papers only treat a mathematical model in computer
simulation. In these terms such papers are not complete.
They have lost the solid ground of physics.
There is of course a reason for that:
"Chaos" implies that even the slightest error in the
initial conditions will have tremendous effect on
the outcome of an experiment. How can you then compare
simulation and experiment, where errors are unavoidable?
Some systems can not experimentally dealt with: e.g. the weather.

6. Quantum Physics (Q.P.): Q.P. is maybe the most successful
theory of physics. Today it is assumed that the fundamental
things have to do with Q.P. Q.P. brings back the integer
numbers into physics, after ~300 years of rule of the real
numbers. Eigen-states, energy states etc etc can be numbered.
They are discrete. This theory of the world can not be 
combined with the continuum of real numbers, D.E. 
and consequently "Chaos Theory". So far both fields
of science have nothing in common and the question arises
if "chaos" will collapse if integer numbers will rule again.


This is in brief a  incomplete and sometimes certainly incorrect
summary some of Mr. Wehr's
argumentation. I'm not an expert in these questions by any means,
but #1 and #5 did really impress me. I can see the problem
that's lying behind.


m.c.