"introducing Chaos" Book Report

Math 12 Book Review Assignment

Introducing Chaos Math 12 Book Review Assignment Meng Ye. Ren June 1, 2009

Book Content Summary.........................................2 Personal Response................................................4 Bibliography.........................................................7 Meng Ye. Ren • Email : [email protected] • Lord Byng Secondary School

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Math 12 Book Review Assignment

Book Content Summary In mathematics, chaos theory describes the behavior of certain dynamical systems – that is, systems whose states evolve with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect).

“Introducing Chaos” is written by Ziauddin Sardar and Iwona Abrams in 1999. It is one of the “Introducing” serie. The whole book is constructed with large number of illustrations. With one big picture on every page and several lines of words, this book is written for the readers with a zero start on this topic. Although it is not a professional scientific researching book, it is clearly enough and give readers a deep and insight recognition on the chaos theory. Next, I am going to brief words to introduce several main points that give me ideas. What is Chaos? I believe most of us have heard about “A butterfly flapping its wings in Brazil will possibly cause a tornado in Texas. ” But still, one thing is needed to be clarified that none of the tornados has been proved to be caused by a butterfly, and that statement is only theoretically possible is because right now we are unable to observe every single butterfly and their impact on climate. The book explains what is chaos and the necessity of existence of this field of science. In our daily life, a little change on something’s initial condition will change the result in a surprising way, and the behavior of which appears to be random to us. A chinese ballad tells us such a dramatical story:“A nail lost, a horseshoe spoiled; A horseshoe spoiled, a horse’s leg broke; A horse’s leg broke, a knight wounded; A knight wounded, a battle lost; A battle lost, an empire ended.” As in the story, the lost of a small nail causes the subjugation of a huge empire. Although the result sounds ridiculous, the development of things make good reason. Those examples show that we all live in a world of chaos: a tiny change on the initial point may give a huge impact on the result.

Meng Ye. Ren • Email : [email protected] • Lord Byng Secondary School

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Math 12 Book Review Assignment

The chaos theory is applied in many scientific disciplines. It is also vastly used in predictions. Nowadays, human beings are dependent on predictions, the weather forecast, the economy, the population dynamics. Such predictions cannot be linear functions, and thus chaos theory is involved. Attractors Attractor is a very important component in a chaos system. Attractors represent the states to which the system eventually settles, depending on the properties of the system. The book uses a very vivide example to explain it. It tells us to imagine a marble swirling around a bowl. The marble eventually settles at the bottom of the bowl. The point at which the marble settles attracts the marble. (45) Strange attractor is a kind of attractor that has special features. Itself is chaotic and unpredictable. For example if you throw a ball to the ocean, it will start a serie of motion, and eventually flow on the water surface. So the water surface is a attractor, and because itself is not static, it is a strange attractor. The concept of attractor explains us the chaotic mechanism more clearly. Other Points in the Book Despite explaining the chaotic mechanism, this book use half of the content introducing the application of chaos theory in every field including biology, Quantum theory, fractal geometry, economics, management, weather forecasting, city planning, architecture, or even religion. It is a introductory book, and on that point it succeeded, because it provide readers with a comprehensive overview on chaos theory. And it also arouses interest of readers to make deeper research on some specific aspect.

Meng Ye. Ren • Email : [email protected] • Lord Byng Secondary School

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Math 12 Book Review Assignment

Personal Response Reflection on the Book Although chaos theory is a newly developed and high-end science, to the book itself, it is not hard to understand for a Grade 12 students. The language doesn’t contain too much scientific terms and the book is written with huge number of examples to make it more suitable for all kinds of readers. And we don’t expect a Grade 12 student to get the whole idea of the chaos theory, while this book may provide a new view of the world to students. The world and the nature is made of complexity. Few things keeps in a constant or linear state, which is different from what we learned in some easy math and physics that for a great many times we use an ideal model. Chaos makes it more practical and realistic. Compare to the theoretical ideal models, chaos has more developing room for people nowadays. Further Research In order to observe the genetic change in human population, I made a program modelling this process.The genetic model is that everyone has two genes on A property. If either of the genes is dominant, this person owns A property, otherwise he doesn’t have A property. The program is basically made of two parts. One is to control the human population, which I give an initial of 100,000, and the other is to match two people and determine their child’s gene property. Still, there are two initial condition I need to input. One is the initial ratio of dominant genes (let’s call it “1”) and recessive genes (let’s call it “0”) in the entire population, and the other is the initial ratio of “10” and “11” in the A population, because these factors also affect the result. This program is written in Java. The original script is available here: http://www.mengyer.com/project/sord.zip First of all, I input initial condition of the data I collected from the internet, which shows the situation of our real world well. Meng Ye. Ren • Email : [email protected] • Lord Byng Secondary School

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Math 12 Book Review Assignment

Chart 1 130000 123500 117000 110500 104000 97500

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Population

Tendency

I printed out a 400-year prediction in my program, but in those chart I only printed out 100 years. If you want to see the whole original data of the 400-year prediction, please visit: http://www.mengyer.com/project/400yrs.html

Chart 2 70.0

52.5

35.0

17.5

0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Ratio (non A%)

inner Ratio (Dominant%)

Meng Ye. Ren • Email : [email protected] • Lord Byng Secondary School

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Math 12 Book Review Assignment

We can clearly see that the population tends to be stable around 120,000, and the percentage of non A is already stable from the very beginning (because that is the real data.). And the “inner Ratio”, which I gave it a “60%” as a initial condition, keeps dropping, and in my data, it comes to stable around 36%. That is almost the similar result given the same initial condition(Ratio 37%, inner Ratio 60%). Then let’s try some other. If I input that non-A is 90% in the initial point, interestingly, during the 400-year prediction, the Ratio of non A% keeps dropping to 37%, and due to the computing ability of my computer, I can’t know if it comes to stable in the next years. In another experiment, it rises up to 93%. This shows the unpredictability of it in some initial conditions. It is a complex chaos system. I use a non-linear equation to control my birth rate and death rate and also the total population and I use randomness to generate genes in the designed ratio. I can make two conclusion from my researching. First, I found the attractors of my model, and those attractors make the model in a stable status.The population settled to a certain range, and the result that we are looking for, the non-A percentage also shows this characteristic. Secondly, the result is largely affected by initial condition. Although my change on the initial data is too big, the experiment still well reflect this feature. The result becomes extremely unstable and unpredictable. It is like change a little condition such as supposing the pigs can fly, can then the whole world would get into a total disorder. This experiment improves my understanding on some aspects of chaos system explained in the book. Social Significance As I have already mentioned in the content summary, the chaos theory is now applied to most of the industries. In astronomy, chaos let people recognize that our universe is not a stable object. Our Earth’s magnetic polar reverse is also a direct result of its internal chaos. Doctor are also using chaos theory to expound the electroencephalogram, which is providing us new information on the researching work on human brains. Meng Ye. Ren • Email : [email protected] • Lord Byng Secondary School

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Math 12 Book Review Assignment

Global warming is a topic of global concern. Climatologist are trying to predict the climate for ten years to fifty years in the future. How can they make such a work? They need to construct a huge model. The chaos theory helps them to avoid unnecessary mistakes and make the result more accurate. Mathematical Relation & Application To the mathematical field, the chaos theory plays an extremely crucial rule. One example mentioned in the book is the water tap.(57) When water is running out of the tap, this seemingly regular motion is also chaotic. Scientists have been trying to measure the time interval of the water drops in order to predict all the system. But, soon scientists discovered that it is impossible, because every measuring has errors, and errors are magnified every time. In this chaotic system, errors will eventually become inconceivable mistakes. Fractal geometry is involved in those systems as well. It, along with chaos theory and Quantum theory has been three top active area of mathematical researching since 1994. The book talks an example about measuring the coastline of England, which is irregular in shape.(32) How small would the minimum measuring unit be? The length will approaching infinite as we decreasing the unit. And then the book actually talks about “fractal dimension” that solves the problem. These are all examples that chaos theory is largely relevant to mathematics. Simple Conclusion From reading “ Introducing Chaos”, I obtain lots of ideas, and even want to continue on this subject. It is a amazing book, and absolutely worth your reading.

Bibliography Sardar, Ziauddin, and Iwona Abrams. Introducing Chaos. Australia: Totem Books, 1999. Print. "Chaos theory." Wikipedia. Web.31 May 2009. . "混沌学 (Chaos)." 百度百科 (Baidu Encyclopedia). Web.31 May 2009. . Meng Ye. Ren • Email : [email protected] • Lord Byng Secondary School

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