Chaos: A short Introduction

Scientists have been coming up with theories for thousands of years now. The theories have been tried and tested have been altered and improved to better understand why certain things happen in and out of this world. Many of these theories are still in the works and our not fully comprehendible and scientists have been working hard to achieve breakthroughs in these areas. One such theory has achieved several breakthroughs within the past couple of years and that theory is called Chaos Theory. This theory first started out mainly as a sort of fictitious happening only read about in novels until lately when scientist have discovered that this theory may have more effect than previously deliberated. In Leonard Smith’s book, “Chaos: A Very Short Introduction” he gives an understandable account of what the Chaos Theory is and how it affects the environment around us. Leonard explains to the reader how the Chaos Theory works through mathematical diagrams, models, and simulations to show the possible results the theory could claim on the environment around us. The book is for the most part easy to understand and gives the reader a sufficient understanding even if the reader does not have a background in mathematics or science. Chaos theory first came about when scientist decided they wanted to be able to predict the future more accurately of certain events happening. The theory takes a look at the initial beginning of a system and tries to determine its future behavior. The system is for the most part random yet has a deterministic quality because the initial conditions already lay out the future of the system and is defined by those initial conditions with no chance of randomness. Leonard points out three chaotic mathematical systems and they are; chaotic systems are non-linear, deterministic, and unstable in that they display sensitivity to initial conditions. It is non-linear in the sense that the input to a certain system does not directly result in the output of the same system. In other words the input could cause a much larger output than expected.
As mentioned above chaotic systems are non-linear, do not have an exact pattern to their system and, can be of exponential growth. To more fully understand the concept of non-linearity in a chaotic system it is pertinent to understand the differences between a linear system and a non-linear system. Take for example an average month that consists of thirty days. Starting out on the first day you are paid a penny; on the second day you would be paid two pennies, and on the third you would be paid three pennies and so on until you are paid thirty pennies on the last day of the month. As you can see the system is relatively stable and does not increase very rapidly. This type of system is a linear system. On the other hand if you take that same thirty day month and take one penny on the first day and double each day after that for the thirty days the amount you would end up with at the end of the month would be substantially greater than that of the first linear system. This is an example of exponential growth that could take place in a chaotic system. Another example of exponential growth used in finance is compounding interest. Many professionals involved in the finance sector use a system called the rule of 72. The rule of 72 can be used to find out how long it will take an amount of money to double given a percentage of increase and an initial amount of principal. The rule of 72 is helpful for many classes of people, but is mostly used with the wealthy when they are interested to see how much money and how long it will take to double their wealth. To better understand how this rule works a practical example will be useful. A teenager is trying to save for college and is wondering how much he will need to put away in a compounding interest account to have enough money to pay for the first year of his college education. For generalization purposes the cost of attendance for the first year is $3,000 dollars. The teenager is 16 and is planning on attending college at 20 years old. The rule of 72 will tell you exactly how long it will take a current investment at a certain percentage rate to double. The teenager was lucky and found a compounding interest account at the rate of 10% compounding annually. The question he asks himself is, how much he will need to invest initially to have the tuition money needed for his first year of college? The rule of 72 says for a 10% interest rate it will take 14.4 years for his investment to double. So how much will his initial investment be so he will have the $3,000? His initial payment into the account would be around $2,000 dollars and in 4 years the amount will grow into the $3,000 needed for tuition. This is done by taking the initial and multiplying it by the percentage and at the end of the year the interest earned will be added to the principal to earn interest as well during the next year. In a linear system you would only receive interest each year on the initial investment and would not receive any interest on the earned interest. As you can see in a chaotic system the numbers will grow exponentially faster even though it is relatively easy to predict the amount of money you will have at a certain date. Tolstoy said it best regarding linear and non-linear systems when he said, “All linear systems resemble one another, each nonlinear system is nonlinear in its own way.”
When you hear the word chaos it gives the impression that it is unpredictable, nothing that happens makes any sense, and it is uncontrollable. However, Chaos theory is a little different from that. Under Leonard mathematical systems of chaos he says that chaos is deterministic in nature. What exactly does deterministic mean? Deterministic in this sense means that everything that happens is directly related to prior occurrences. In other words it just doesn’t happen out of thin air everything that takes place was determined through other events that lead up to the current event. It shows that there is a sense of infiniteness to the possibilities of a chaotic system, the events could continue to happen leading to other events and so on. Take for an example John meets Jane. In deterministic view if you see into the future and find that John and Jane are married you could say that all the events that led up to the marriage were essential to them getting married. It could even be said that there was a linking of certain events that had to happen in a certain order for it to lead up to the actual marriage. Then you could even go further and say that Larry, Curly, and Moe were born during these chains of events after John and Jane were married. Later; Larry’s son, Harry, becomes president of the United States. Harry comes out with a fabulous fiscal policy and saves the economy from going under. The economy being saved was determined long ago when Larry’s parents met and conceived him through a series of events that had to happen for Harry to have been able to save the economy. These events could continue on forever and each event in the chain was determined by the previous events. It is mind boggling to think about this, and can be very hard to comprehend the magnitude that one small event could have such a large impact.
There is more to predicting a chaotic system than most people realize. Predictions can be made and are more likely to be right in the short term than in the long term. The underlying factor in how accurate a chaotic system can be is the level of exactness of those initial beginnings of the system. If the initial beginnings of the system have a low margin of error then the further ahead a system will be predictable. The problem that scientist and mathematicians run into is that it is hard to know exactly the initial beginnings of a system to be able to accurately predict what will happen in the system. Take, for example, a company that you might be interested in investing. If you invest in this company without looking at the balance sheets, financial statements, and history; you will not have all the necessary information to provide a good prediction of where the company’s stock will be headed. On the other hand if you do the hard work and study up on the company checking its financial statements, past stock sales, stability of the company and anything else pertinent to making a decision you will find that you will have a low margin of error and will be able to predict more accurately where the stock is headed in the future. In the finance and economics world professionals are always developing new models that will better predict where the market is moving to. These models work on feeding off the initial information to predict what may happen in the future and the more accurate the information the better the chance that the model has to predict the system’s future. A problem that makes these types of systems so difficult to predict is that in reality, the world we live in, anything can happen and anything can change at a drop of a hat. One day your company’s stock could be going off the charts and the next day a scandal like Enron and other companies could show up leaving your stock worthless. That is why the initial beginnings of the system are so important to have as exact as they can be.
In closing Chaos theory is all around us and is becoming more and more prevalent in our daily lives. When you look at weather forecasting it is by far perfect, but these chaotic systems that are used to predict the weather are improving every year. Predictions of weather phenomena and other things such as earthquakes and tsunamis are becoming increasingly more accurate and potentially could save thousands of lives in the future. Predicting has been a large part of our society, from prophets of the bible to people like Nostradamus to those working the stock market. If you understand chaotic theory and are able to apply it to certain systems in your life you will be able to predict future outcomes easier than you would without the knowledge of this fantastic theory. It is possible that one day scientist will understand Chaos more completely and perhaps be able to predict the future and perhaps change possible undesirable outcomes.

References
Smith, L.A. (2007). Chaos: A Very Short Introduction. New York: Oxford University Press.