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Dynamical Systems
A dynamical system is any process that moves or changes in time. Dynamical systems occur in every branch of science. For example: the motion of the planets, the weather, the stock market, and finally chemical reactions.
The motions of the planets in celestial mechanics are a good example of a process of something that evolving in time. The weather is another system that changes dramatically over time.
Similarly, the Stock Market, economic systems are good examples of very chaotic at times, dynamical systems. Finally, in chemistry, simple chemical reactions are examples of processes that evolve in time.
Can you predict what will happen?
When a scientist confronts a dynamical system, the question that she or he ask is can I predict what will happen in the future, Can I predict how this motion will evolve in time? If you look at some of the examples giving of dynamical systems, it is clear that some of them are predictable.
The motion of the planets for example; you know that in the morning when you wake up the sun will rise. Similarly, chemical reactions, you know that tomorrow morning when you put crème in your coffee, the resulting chemical reaction will not be an explosion.
On the other hand, the weather or the stock market, those are examples of dynamical systems that seem to be unpredictable. The question now, is why are they so unpredictable?
A person might say I know why the weather and stock market are so unpredictable. Those are dynamical systems that seem to depend on so many variables, that it would be impossible for anyone to know all of the variables at any one time so to make a prediction.
For example to predict the weather, you would have to know all elements of the weather around the globe instantaneously. You would have to know the barometric pressure, the wind speed and direction everywhere in the globe in order to predict what the weather will be like a week, hence.
The main maxim of science is its ability to relate to cause and effect. On the basis of the laws of gravitation, for example, astronomical events such as eclipses and the appearances of comets can be predicted thousands of years in advance.
Other natural phenomena, however, appear to be much more difficult to predict. Although the movements of the atmosphere, for example, obey the laws of physics just as much s the movements of the planets do, weather prediction is still rather problematic.
We speak of the unpredictable aspects of weather just as if we were talking about rolling dice or letting an air balloon loose to observe its erratic path as the air is ejected. Since there is no clear relation between cause and effect, such phenomena are said to have random elements.
Yet there was little reason to doubt that precise predictability could, in principle, be achieved. It was assumed that it was only necessary to gather and process greater quantities of more precise information (e.g., through the use of denser networks of weather stations and more powerful computers dedicated solely to weather analysis).
Some of the first conclusions of chaos theory, however, have recently altered this viewpoint. Simple deterministic systems with only a few elements can generate random behavior, and that randomness is fundamental; gathering more information does not make it disappear. This fundamental randomness has come to be called chaos.
Similarly, the stock market, to predict where the stock market will go, theoretically at least, you have to know the behavior of all elements of the economy, of all the consumers in the economy. Clearly these are systems that depend on just too many variables to make a prediction.
Well, it is certainly true that the weather and the stock market depend on too many variables, but on the other hand, that’s not necessarily the reason that makes these dynamical systems unpredictable.
It’s one of the most remarkable discoveries of mathematicians in the last thirty years that very simple dynamical systems, systems that depend on only one variable, not billions of variables like meteorology or the economy,
systems that depend on a single variable can behave just as unpredictably, just as turbulently as the weather or the stock market. That’s what will be discussed more, is how the simple dynamical systems can react or behave in a very strange and chaotic way.
An evident inconsistency is that chaos is deterministic, generated by fixed rules, which do not themselves involve any elements of change (Springer – Verlag 11). People even talk about deterministic chaos (11).
In principle, the future is completely determined by the past; but in small doubts, much like minute errors of measurement, which enter into calculations, is amplified, with the effect that even though the behavior is predictable in the short term, it is unpredictable over the long run (11).
A landmark achievement of tremendous, accelerating effect was made about three hundred years ago with the development of calculus by Sir Isaac Newton (1643 – 1727) and Gottfried Wilhelm Freiherr von Leibniz (1646 – 1716) (11).
Through the universal mathematical ideas of calculus, the basis was given with which to they say that successfully model the laws of the movements of planets with as much aspect as that in the development of populations,
the spread of sound through gases, the conduction of heat in media, the interface of magnetism and electricity, or even the path of weather events (11).
Also growing during that time was the secret belief that the terms determinism and predictability were equivalent (11- 12). Present, past, and the future are joined together by casual relationships;
and along with the views of determinists, the problem of an exact prognosis is only a matter of the difficulty of documenting all the relevant data (12).
In addition, chaos and order, specifically the causality principle, can be observed in coincidence within the same system (12). There may be a linear progression of errors characterizing a deterministic system, which is controlled by the causality principle (12).
While, in the same system, there can also be an exponential chain of errors, for example the butterfly effect, indicating that the causality principle breaks down (12).
In other words, one of the lessons coming out of chaos theory is that the soundness of the causality principle is narrowed by the uncertainty principle from one end as well as by the inherent properties of fundamental natural laws from the other end (12).
SOLAR SYSTEM CHAOS
Chaos theory isn't new to astronomers. Most have long known that the solar system does not "run with the precision of a Swiss watch." Astronomers have uncovered certain kinds of instabilities that occur throughout the solar system in the motions of Saturn's moon Hyperion, in gaps in the asteroid belt between Mars and Jupiter, and in the orbits of the system's planets themselves.
As used by astronomers, the word chaos denotes an abrupt change in some property of an object's orbit. An object behaving in a chaotic manner may, for example, have an orbital eccentricity that varies cyclically within certain limits for thousands or even millions of years, and then abruptly its pattern of variation changes.
The result is a sharp break in the object's history -- its past behavior no longer says anything about its long-term future behavior. For centuries astronomers have tried to compare the solar system to a gigantic clock around the sun.
But they found that their equations never actually predicted the real planets' movement. This problem arises from two points, one theoretical, and the other, practical.
The theoretical difficulty was summed up by Henri Poincare around the turn of this century. He demonstrated that while astronomers can easily predict how any two bodies -- Earth and the Moon, for example -- will travel around their common center of gravity, introducing a third gravitating body (such as another planet or the Sun) prevents a definitive analytical solution to the equations of motion.
This makes the long-term evolution of the system impossible, in principle, to predict. The practical difficulties are the limits of computer power. Even with the help of calculators and desktop computers, the long-term calculations were too lengthy.
The conclusion from all this is that while new real-life chaos discoveries are being made, current computing technology cannot keep up with the pace.
CHAOS AND THE STOCK MARKET
According to respected authorities, stock markets are non-linear, dynamic systems. Chaos theory is the mathematics of studying such non-linear, dynamic systems. Chaos analysis has determined that market prices are highly random, but with a trend.
The amount of the trend varies from market to market and from time frame to time frame. A concept involved in chaotic systems is fractals. Fractals are objects that are "self-similar" in the sense that the individual parts are related to the whole.
A popular example of this is a tree. While the branches get smaller and smaller each is similar in structure to the larger branches and the tree as a whole. Similarly, in market price action, as you look at monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance.
Just as with natural objects, as you move in closer and closer, you see more and more detail. Another characteristic of chaotic markets is called "sensitive dependence on initial conditions." This is what makes dynamic market systems so difficult to predict.
Because we cannot accurately describe the current situation band because errors in the description are hard to find due to the system's overall complexity, accurate predictions become impossible.
Even if we could predict tomorrow's stock market change exactly (which we can't), we would still have zero accuracy trying to predict only twenty days ahead.
A number of thoughtful traders and experts have suggested that those trading with intra day data such as five-minute bar charts are trading random noise and thus wasting their time.
Over time, they are doomed to failure by the costs of trading. At the same time these experts say that longer-term price action is not random. Traders can succeed trading from daily or weekly charts if they follow trends.
The question naturally arises how can short-term data be random and longer-term data not be in the same market? If short-term (random) data accumulates to form long-term data, wouldn't that also have to be random?
As it turns out, such a paradox can exist. A system can be random in the short-term and deterministic in the long term.
Mathematical Dynamical Systems
To simplify the situation, let’s begin by discussing mathematical dynamical systems, a very simple abstraction of the kinds of dynamical systems that arise in nature.
What’s a mathematical dynamical system? Well, among the simplest mathematical dynamical systems are the so-called Iterated “functions.”
Start with any mathematical expression, for example the square root function and start with any number, say x. How does one create a mathematical dynamical system? Well, through the process of iterating this mathematical system.
That is accomplished by taking the initial number x and computing its square root, you get a new number. Then take that number and compute its square root, you’ll get another new number and so forth.
This is the process of iteration. It’s a dynamical system. The numbers are changing in time. The question to the mathematician is, just as in the case of the scientist, “Can you predict what will happen?”
“Can you predict what will happen when you iterate this function over and over again?”
Now one can easily see that this process is well suited for the use of a computer. There’s nothing a computer can do better than iterate functions over and over again (VIDEO).
An iterator comes with a bunch of numbers that you can input together with a bunch of functions that you can iterate. What functions the iterator has is up to whom ever programs the computer or calculator.
How does one do the process of iteration? Well, for the simple minded you can use a calculator for the first few examples. Well, with the square root example, start by imputing your favorite number into the calculator.
For example, you might put in the number 256. You then iterate the square root function by pressing the square root button, then computing the square root of 256. The answer is 16.
To iterate, you would just do it again. The square root of 16 is 4 and the square of 4 is 2. The square root of 2 is 1,41… Then you may ask, “What happens when we do this over and over again?”
Iterate the square root of 1.41… Keep hitting the square root button and eventually you’ll see that no matter what number you started with, you’ll always end up with the number 1.
That is an excellent example of an iterated process that is completely predictable. No matter what number you start with on the square root function, you always end up with the number 1.
Here’s another example:
Let’s take the function x^2. Start with any number. Let’s start with the number 2. When you iterate the squaring function you first get 4. Square 4, you get 16, square 16 and get 256.
You can see what happens. Square 256 and you get 256^2, a rather large number. You see that upon iteration, repeated squaring when you start with any number greater than 1, it tends to infinity.
Once again, that’s an example of an iterated function whose behavior is completely predictable.
Here’s another example:
Take the sin function. What happens when you iterate the sin function? Well, start with any number, say 123 and iterating the sin of 123, you get -.45… sin of that is -.43… Iterate the sin again and you get -.41…
And you see what happens. Iterating the sin function over and over again, eventually yields after 300 or more iterations the number 0. So if you iterate the sin function, no matter what number you start with, you always end up with the number 0.
Again, a perfectly predictable iterated process.
Another example:
Instead of using the sin function, use the cos function. What happens when you iterate the cos function? Well let’s see. Start with any number, say 123 and what do you think happens when you iterate the cos function?
What happens when you iterate cos over and over again? It turns out that with no matter what number you start with, when you iterate cos in radians, you always end up with the number .73908…
Where did that number come from? That will be discussed later on, but for now notice that you know how to iterate or what the result of the iteration of cos will be. No matter what number you start with, with cos you always end up with that strange number.
There are many dynamical systems that can produce chaos. However, the focus will now be on only one particular transformation. It is the quadratic transformation, which comes in different forms, for example, x -> ax(1 – x).
How about iterating the quadratic function 4x(1 – x). What happens when you iterate this simple quadratic expression? Well let’s start with any number. Say the number .4 and what happens when you compute this quadratic expression?
Now if you plug in .4 into the quadratic expression, you get .96. Now iterate again and you get .154. Iterate again and you get .521 and iterate again and you get .998. Iterate yet another time and you get .008. Iterate once more and the result is .032. See the pattern?
Try some more iterating. Iterate again and the result you get will be .123. Iterate yet another time and you get .431. Do it again and get .980, once more and the answer is .078. See the pattern? Probably not. Iterate again and you get .288, once more and you get .823.
There is no pattern whatsoever when you iterate the quadratic expression, because this expression is Chaotic, totally unpredictable. For all intense and purpose, iterating 4x(1 – x) is a random number generator.
Now most iterators or calculators don’t come with a 4x(1 – x) button, but that’s no problem. On most computers you can easily program it to iterate a quadratic expression.""
