[sdiy] slightly ot: Chua circuit and chaos

[Richard Wentk]

At 13:02 11/02/2003 +0100, Czech Martin wrote:
>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?

This will be covered by any decent university comp sci course. It's usually 
called something like 'computing with approximate numbers.'

You can't learn scientific programming to degree level without being aware 
of the limitations of only having a certain level of accuracy. That's 
certainly true for DSP programming as well as other things.

>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.

Intuitively I wouldn't find this surprising for small integer values, 
because all you're really doing is sacrificing accuracy.

If you ran the same simulation with really really really huge integer 
values I'd expect chaotic effects to reappear again. By using a narrow 
value range you're effectively quantising the chaos out of existence.

>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.

Well, kind of. In practice most differential equations, especially second 
order and higher, can't be solved analytically. So they're solved 
numerically instead. And once you do that you're back in the world of 
computing with approximate numbers again. People who work with these things 
do understand that solutions are approximate, and there are various ways to 
estimate how accurate a solution is likely to be.

>3. "Chaos Theory": "Chaos Theory" as such doesn't exist.

I think you'll find this isn't true. :)

>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.

This I'd agree with.

But then most people don't have the background to understand the underlying 
maths. That's why you get nice stories like the one about the butterfly 
causing a storm. It's not good science, but it gets a story of sorts across 
that's kind of relevant, in a not very relevant sort of way. ;)

>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.

But then it becomes a question of providing a horizon of predictability 
that's good enough. Most people don't care about predicting the exact 
weather a hundred years from now. Being able to predict what happens 
tomorrow is going to be enough. And as long as your dataset is complete 
enough, you can still do that.

>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.

In a fuzzy kind of way. :)

The probability density functions that describe electron shells are most 
definitely continuous, not discrete. The discreteness comes from the way 
that different density functions are clearly distinct, and an electron can 
only be described by using one function, and not two, or some interpolated 
combination.

Richard