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- Herausgeber: Icon Books
- Kategorie: Wissenschaft und neue Technologien
- Serie: Graphic Guides
- Sprache: Englisch
If a butterfly flaps its wings in Brazil, does it cause a tornado in Texas? Chaos theory attempts to answer such baffling questions. The discovery of randomness in apparently predictable physical systems has evolved into a science that declares the universe to be far more unpredictable than we have ever imagined. Introducing Chaos explains how chaos makes its presence felt in events from the fluctuation of animal populations to the ups and downs of the stock market. It also examines the roots of chaos in modern maths and physics, and explores the relationship between chaos and complexity, the unifying theory which suggests that all complex systems evolve from a few simple rules. This is an accessible introduction to an astonishing and controversial theory.
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Veröffentlichungsjahr: 2014
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Published by Icon Books Ltd, Omnibus Business Centre, 39-41 North Road, London N7 9DP email: [email protected]
ISBN: 978-184831-766-6
Text and illustrations copyright © 2013 Icon Books Ltd
The author and artist have asserted their moral rights.
Originating editor: Richard Appignanesi
No part of this book may be reproduced in any form, or by any means, without prior permission in writing from the publisher.
Contents
Cover
Title Page
Copyright
Yin, Yang and Chaos
Ancient Chaos
Chaos Theory
Why is Chaos Exciting?
Where Does Chaos Come From?
Defining Chaos
The Language of Chaos Dynamic, Change and Variable
Systems
Defining Systems
Periodic and Aperiodic Equations
What is Unstable Aperiodic Behaviour?
Linear Systems
Nonlinear Complication
Feedback
The Three Body Problem
Chaos Modelling
Questions of Long-Term Behaviour
The Signature of Chaos
The Little Devil
Benoit Mandelbrot and Fractal Geometry
Chaos and Order in Economics
Chaos on the Telephone Lines
Measuring the Coast
Fractal Dimensionality
What are Fractals?
Fractals are Everywhere ...
The Julia Set
The Use of Fractals
Edward Lorenz
Small Differences, Big Consequences
The Water Wheel Example
Strange Attractors
Cultural and Identity Attractors
Chaotic Attractors
Representing Phase Space
Phase space makes a dynamic system easy to watch.
The Lorenz Attractor
The Butterfly Effect
David Ruelle
What is Turbulence?
How Does Turbulence Happen?
Ruelle’s Approach
Robert May and Animal Populations
May’s Bifurcations
Chaos in Real-Life Events
Mitchell Feigenbaum: Nonlinear Patterns
Easy Solutions to Difficult Problems
Ilya Prigogine: Dissipative Systems
Disorder to Order
Self-Organization and Time
Time and the Problem of Entropy
The Source of Order
Other Features of Self-Organization
Period Three Chaos
Towards the Edge of Chaos: Complexity Theory
What is Complexity?
Adapt and Relate
Beyond Entropy
Chaotics
Chaos and Cosmos
Poincaré’s Discovery
The Conditions of Stability
Quasi-Periodic Stability
The KAM Theorem
Saturn’s Moons
A Chaotic Universe
Quantum Chaos
Brief History of Quantum Theory
The Black Body Problem
Applying Planck’s Constant
Probability Waves
Chaos in Quantum Physics
Chaos In Between States
Chaos and Economics
Feedback in Economics
The Problems with Equilibrium
Increasing Returns in High-Tech
Beware of “Initial Conditions”
The End of Neo-Classical Economics
How to Play Monopoly
Chaotic Management
Anticipating Future Breakthroughs
Enablement and Forecasting
Chaos and Cities
Fractal Cities
Fractal Skylines
Dissipative Cities
Local and Global Chaos
Control or Participation
Chaotic Architecture
Chaos and the Body
Body Fractals
The Heart’s Attractor
Chaos in the Heart
Chaos and Good Health
Chaos and the Brain
A Chaos Model of Consciousness
Chaos and Weather
Long-Term Weather Prediction
What About the Greenhouse Effect?
Chaos and Nature
Scientific Safety
The New Nature
Is It Safe?
Post-Normal Science
Chaos and the Non-West
Criticism of Chaos
Further Reading
Biographies
Acknowledgements
Index
Yin, Yang and Chaos
Ancient Chinese thought recognized that chaos and order are related. In Chinese myth, the dragon represents the principle of order, yang, which emerges from chaos. In some Chinese creation stories, a ray of pure light, yin, emerges out of chaos and builds the sky. Yin and yang, the female and male principles, act to create the universe. But even after they have emerged from chaos, yin and yang still retain the qualities of chaos. Too much of either brings back chaos.
Ancient Chaos
Hesiod, a Greek of the 8th century B.C., wrote the Theogony, a cosmological poem which states that “first of all Chaos came to be”, and then the Earth and everything stable. The ancient Greeks seem to have accepted that chaos precedes order, in other words, that order comes from disorder.
Nothing further was made of this ‘mythical’ idea ...
Until recently in the 20th century when chaos theory arrived.
Chaos Theory
Chaos theory is a new and exciting field of scientific inquiry.
The phenomenon of chaos is an astounding and controversial discovery that most respectable scientists would have dismissed as fantasy just a decade or so ago.
But today it is seen as one of the most notable since the advent of quantum theory in the early 1900s.
If chaos theory fulfils its potential, it will dramatically change the way we view the natural world and ourselves.
Why is Chaos Exciting?
Chaos is exciting for all these reasons ...
It connects our everyday experiences to the laws of nature by revealing the subtle relationships between simplicity and complexity and between orderliness and randomness.
It presents a universe that is at once deterministic and obeys the fundamental physical laws, but is capable of disorder, complexity and unpredictability.
It shows that predictability is a rare phenomenon operating only within the constraints that science has filtered out from the rich diversity of our complex world.
It opens up the possibility of simplifying complicated phenomena.
It combines imaginative mathematics with the awesome processing power of modern computers.
It casts doubt on the traditional model-building procedures of science.
It shows that there are inherent limits to our understanding and predicting the future at all levels of complexity.
It is strikingly beautiful! Shakespeare had it right when he had Hamlet say in Act 1, scene 5 ...
There are more things in heaven and earth Horatio, Than are dreamt of in your philosophy.
Hi! I’m Cordiallia Cauliflower. Just look at what chaos has done to me!
Where Does Chaos Come From?
Three major recent developments have made chaos a household word.
1. Breathtaking computing power that enables researchers to perform hundreds of millions of complicated calculations in matters of seconds.
2. The rise in computing power has been accompanied by a growing scientific interest in irregular phenomena such as ...
random changes in weather
the spread of epidemics
the metabolism of cells
the changing populations if insects and birds
the rise and fall of civilizations
the propagation of impulses along our nerves
3. Chaos theory was born when these developments were combined with the emergence of a new style of geometrical mathematics ...
Beyond the familiar shapes of Euclidean geometry ...
To non-Euclidean structures of fractal geometry.
These developments have made an impact in almost every field of human endeavour. Chaos theory has been like a sea into which flow the rivers and tributaries of almost every discipline and subject – from mathematics, physics, astronomy, meteorology, biology, chemistry, medicine to economics and engineering, from the study of fluids and electrical circuits to the study of stock markets and civilizations.
Defining Chaos
Chaos has been variously defined. Here are just a few examples ...
“A kind of order without periodicity.”
“Apparently random recurrent behaviour in a simple deterministic (clock-work-like) system.”
“The qualitative study of unstable aperiodic behaviour in deterministic nonlinear dynamical systems.”
And here’s another by a mathematician in the field, Ian Stewart.
The ability of simple models without inbuilt random features, to generate highly irregular behaviour.
Technical definitions of chaos are not easy to understand. So let’s begin to familiarize ourselves with its terminology.
The Language of Chaos Dynamic, Change and Variable
Chaos is a dynamic phenomenon. It occurs when something changes. Basically, there are two types of changes.
Regular ones studied by classical physics and dynamics.
And chaotic ones. There may be other types which we have not discovered yet!
What is changeable in a given situation is referred to as a variable.
Systems
Any entity that changes with time is called a system. Systems thus have variables. Here are some examples of systems.
The human body
The population of penguins in the Antarctic
Molecules in an imaginary box
Flu moving through a country
‘The X Flies’
A school
Change is inevitable, except from a vending machine.
Defining Systems
A deterministic system is one that is predictable, stable and completely knowable. The classic example of a deterministic system is an old-fashioned grandfather clock. The balls on a snooker table behave within the boundaries of a deterministic system.
In classical physics, the universe itself was considered to be a deterministic system.
Give me the past and present co-ordinates of any system and I will tell you its future.
In linear systems, variables are simply and directly related. Mathematically, a linear relationship can be expressed as a simple equation where the variables involved appear only to the power of one:
There are no squares, cubes, fourth powers, etc. These types of equations can be solved easily, even if they involve several variables.
Nonlinear relationships involve powers other than one. Here is a nonlinear equation:
Such equations are much harder to analyze and frequently need the help of a computer to understand.
Periodic and Aperiodic Equations
A period is an interval of time characterized by the occurrence of a certain condition or event. A variable in a periodic system exactly repeats its past behaviour after the passage of a fixed interval of time – think of a swinging pendulum.
Aperiodic behaviour occurs when no variable affecting the state of the system undergoes a completely regular repetition of values – visualize the flow of water as it goes down a sink.
Unstable aperiodic behaviour is highly complex. It never repeats itself and continues to show the effects of any small perturbation to the system. This makes exact predictions impossible and produces a series of measurements that appear random.
That’s why, in spite of our satellite observations and computer models, it is still impossible to predict the weather accurately.
What is Unstable Aperiodic Behaviour?
Behaviour that is unstable yet periodic is difficult to imagine – indeed, it appears to be a contradiction in terms. However, human history provides us with several examples of just such a phenomenon. It is possible to chart broad patterns in the rise and fall of civilizations. We can see that these patterns are periodic. But we know that events never actually repeat themselves exactly. In this realistic sense, history is aperiodic. We can also read in history textbooks that seemingly small unimportant events have led to long-lasting changes in the course of human affairs.
Until quite recently, our principal image of behaviour that IS so complex as to be unstable and aperiodic was the image of a crowd.
Now that our perception has changed, we see such behaviour in even the commonest events: water dripping from a tap, a flag waving in the breeze, the fluctuation of animal populations.
Linear Systems
So: simply put, chaos is the occurrence of aperiodic, apparently random events in a deterministic system. In chaos there is order, and in order there lies chaos. The two are more closely connected than we ever thought before.
But since deterministic systems are predictable and stable, this seems to be illogical. As a matter of habit, humans have looked for patterns and linear relations in what they see.
Linear relations allow us to predict what will happen within a system and can easily be expressed on a graph.
In other words, they form a straight line on the graph and we know where that line is going.
Linear relationships and equations are solvable. That makes them easy to think about and work with.
Nonlinear Complication
Nonlinear equations, on the other hand, cannot be solved. Friction, for example, often makes things difficult by introducing nonlinearity. Without friction, the amount of energy required to accelerate an object is expressed in a linear equation ...
Friction complicates things because the amount of energy changes, depending on how fast the object is moving.
Nonlinearity, therefore, changes the deterministic rules within a system and makes it difficult to predict what is going to happen.
1. As the parameter rose, the final population rose slightly too, making a line that rose as it moved from left to right on the graph.
2. Suddenly, as the parameter passed 3, the line broke in two and May had to plot for two populations. This split meant that the population was going from a one-year cycle to a two-year cycle.
3. As the parameter rose further, the number of points doubled again and again. The behaviour was complex yet regular. Beyond a certain point, the graph became totally chaotic – and the graph was completely blacked in. Yet even in the midst of the chaos, stable cycles returned as the parameter was increased.
