Chaos Theory - One dimension Logistic map (Excel)

Chaos Theory - Pretty interesting stuff and somehow very profound. It is yet another of those mathematical models which goes against the very grain of mathematics itself (well the 19th century view of mathematics to be precise).
Mathematics has always been regarded as the holiest of the scientific subjects owing to its inherent properties of being testable, predictable and deterministic. It was natural to assume that if something could be described mathematically then there would remain no surprises and it made perfect sense. Then maths got complicated!
1) Probability Theory - Here, after rigorous mathematical somersaults all you would end up with is the probability of the occurrence of an event not a certainty. Though it can be argued that it does limit the range of possible outcomes and hence is deterministic in some sense.
2) Quantum Theory - My favorite ! The uncertainty principle virtually guarantees that no matter what instrument is used it is impossible to determine the position and momentum simultaneously. Then of course there is the "Measurement problem" the mention of which will send shivers down any Physicists spine. To put it bluntly, if you set up an experiment to measure something like a Photon as a particle then you will detect it as a particle. An experiment set up to detect it as a wave will detect it as a wave. Funny isnt it? :-)
Well thats how the collapsible wave function of gauge boson behaves..deal with it. So essentially here mathematics has determined that you cannot determine.
3) Chaos Theory - Perhaps the most counter intuitive of all theories. Seemingly innocuous linear equations display behavior that is very surprising. I was trying to model the one dimension logistic map equation which is incidentally one of the good example of deterministic chaos. This equation is also very intriguing because this has wide uses in fields as diverse as ecology! ( Population Growth model) . In essence this equation has a very high dependency on the initial conditions and the behavior changes drastically with changes in initial conditions.
The linked spreadsheet has the equation embedded and plots a graph of 200 points based on the input value of the scaling parameter k ( Sheet: Input, Cell: A8). I have put down the ranges of k that induces different behavior from the same equation. Have fun trying it out for yourselves.
Also, Wolframs Mathematica has done a better job of modeling this equation so if you want to try out a widget which allows a greater control of parameters as well as a mean square minimized error then please refer the Wolfram Demonstration Project

And to think, our universe Probably started from Chaotic conditions on a scale that necessitates the existence of Quantum effects.

If God exists he must be ROFLing now.. couldnt have made it more puzzling. :-)

Life - The stochastic model

I have often wondered about the meaning of life and the existential questions. I have always drawn a blank. Big surprise eh? For some reason, I just cannot come to terms with the notion that life just "is" and there is no logical reason or higher purpose behind it. If everything in nature stemmed from a few basics laws then it ought to be possible to bring about some semblance of order in the nature of things. I fully understand the problem of algorithmic compressibility and I am not looking for a way to figure out the future. I have always loved mathematics and all my initial attempts at understanding the universe were inherently mathematical.

I pondered over the fact that cause and effect are interrelated and a finite number of causes should always lead to a finite number of effects. Thus, if it is possible to limit the set of inputs, the output set should also be bounded. Under this hypothesis, if I could narrow down the number of factors that can effect a situation I could hypothetically have a very good chance of finding the outcome. Seemed simple enough but turns out that nature is far more complicated. This hypothesis kept me busy for a while and I attributed my failures to the fact that I was not accounting for all the variables.

Enter Chaos theory. It gave me nightmares. It went completely against my understanding of mathematics but here it was. I coded a computer program on one of the equations and nearly wrecked my machine but it was there and it was undeniable. It just stated that a given input can result in a wide range of outputs which seemed essentially random in nature. My initial hypothesis went out the window. Closely associated with Chaos was the concept of fractals and that puzzled me for a while. My initial consternation gave way to curiosity and I started to feel that a deeper understanding was needed to get to what Chaos was all about. It soon began to seem that Chaotic equations did settle down into a "pattern" after a while and though they do not stabilise, they do remain bounded. There was still hope. If the universe is based on Chaos then it has had enough time to settle into a pattern and we should be able to figure out the bounds.

Then came probability and gate crashed my concepts. The mathematical probability struck me as wondrous. It seemed to say that you can try doing something once and it may work, try twice and it may work but try it a zillion times and you will fail "x" percentage of the time. It was a surprising result but it was true because it is verifiable. Could life indeed be a stochastic process? Modern science does treat it as so and so you will see percentages like children in Somalia have 70% chance of survival etc etc. Statistics such as this are a clear indication that there is a probability lurking around the corner. Probability of survival, probability of success... what else? It seems we are on the verge of a scientific and philosophical breakthrough. A point in our history where we realize that there are just too many decisions to be made and each decision has a certain probability of success. The number of choices we see before us are in turn a result of the choices made earlier. So that's cause and effect. After all this, even in spite of good decisions and sound judgement it is still possible to screw up courtesy Chaos.

Which now makes me wonder about the probability of probability describing life... Hmmm... maybe another day.