## Introduction to Chaos and Nonlinear Dynamics

Authors: Takashi Kanamaru and J. Michael T. Thompson

Japanese

The word "chaos" written in the Japanese characters

Thank you for about 100,000 accesses from 17 Sept. 1997 to June 2005.

## What's New?

 28 April 2013: "Riddled Basin for Android" were created. 28 April 2013: "Mandelbrot Set for Android" and "Fractal Basin Boundary for Android" were updated. 24 December 2011: T. Kanamaru wrote a book titled "Learning Fourier transform with Excel and OpenOffice(in Japanese)". Fourier transform of Lorenz chaos and the sound of Lorenz chaos are explained.

## Contents

• Interactive Dynamic Simulations
• Presentation
• Others

### Interactive dynamic simulations

We present some simulators that demonstrate the well-known chaotic systems.
These interactive simulators will help you to understand the complex properties of nature.

[Platform]
Windows: Java 1.6.0, Microsoft Java VM (Java 1.1.4 compliant)
Linux: Java 1.6.0
Solaris: Java 1.4.2

### Brain Dynamics

The theories of chaos and nonlinear dynamics are applied to many fields such as sociology, economics, and biology.
The one of the most active field is the brain science.

Here we present some simulators which introduce some researches to understand our complex brain.

[Platform]
Windows: Java 1.6.0, Microsoft Java VM (Java 1.1.4 compliant)
Linux: Java 1.6.0
Solaris: Java 1.4.2

### Chaos in Android Smartphone

We offer some applications for android smartphone.

### Chaos Animations

I made some animations of periodically varying chaotic attractor.

### Chaos Gallery

A French mathematician, Henri Poincaré (1854-1912) proved that there is no analytical solution of the dynamical equations governing the three planets system.
He created an original method to understand such systems, and discovered a very complicated dynamics, namely, chaos. He said,

"It is so complicated that I cannot even draw the figure."

Fortunately we are living in the computer age, so we can see chaos in a computer display.

### Quotations about Chaos (Translation from Japanese)

• Scientists
• James Clerk Maxwell "Matter and motion" (1877)
There is a maxim which if often quoted, that "The same causes will always produce the same effects." (…)
There is another maxim (…), which asserts "That like causes produce like effects." (…)
In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a very great change in the final state of the system, (…)

• Henri Poincaré "The science and the hypothesis" (1902)
Are there anything that are more complex than the orbit of planets?

• Henri Poincaré "The value of the science" (1905)
The trajectory of planets is so complex that no one have derived its equation.

• Henri Poincaré "Science and methods" (1908)
An initial small error will result in enormous difference. So no prediction is efficient and statistical phenomena appear.

• van der Pol "Frequency demultiplication" (1927)
Often an irregular noise is heard in the telephone receivers before the frequency jumps to the next lower values.(…)
The shaded parts correspond to those settings of the condenser where an irregular noise is heard.

• Li and York (1975)
Period three implies chaos.

• Hajime MORI "Dissipative structure and chaos" (1994)
With small perturbation, chaos is unstable and not reproducible. But a long-time mean on the path is stable and reproducible.

• Masaya YAMAGUCHI "An introduction to chaos" (1996)
To explain the Oriental philosophy in European language is needed. Chaos is one of these languages. The symmetry causes only periodic motions (like duality), but non-symmetry (a property of Oriental philosophy) causes chaos. Chaos is a source of creation. It is a tradition of the human race before the Greek era.

• Philosophers or Literary Men
• Lao Tzu "The Tao Te Ching" (B.C. the 6th Century)
The Tao produced One; One produced Two; Two produced Three; Three produced All things.

• Lucretius "On the Nature of Things" (About 100 B.C.)
While moving in some direction, at unpredictable times and at unpredictable positions,
it (atom) changes the direction of motion so slightly that the movement can be altered.
Unless the atom often changes the direction of motion obliquely, (…)
the nature could not generate anything.

• Gaius Petronius Arbiter "Satyricon" (A.D. the 1st Century)
Chance has its reasons. (Suam habet fortuna rationem.)

• Blaise Pascal "Pensées" (The 17th Century)
162 (…) Cleopatra's nose: had it been shorter, the whole aspect of the world would have been altered.

By the way, every detail contains accidental elements which are finer than the previous ones or itself, and, to show the reason of each accidental element, similar analyses would be required; therefore, we cannot proceed by such analyses.

• Edgar Allan Poe, "The Gold-Bug" (1843)
of course the error, however trivial in the beginning, increased as we proceeded with the line, and by the time we had gone fifty feet threw us quite off the scent.

• Jules Verne "From the Earth to the Moon" (1869)
"I cannot show you the actual trajectory of the rocket between the moon and the earth. (…)"
"Why?"
"Because this problem is called the three body problem, and the theory of integration has not been developed enough to solve it.

• Jules Verne "Floating Island" (1895)
Surely, even the marvels of science cannot reproduce the beauty of nature.

• Friedrich Nietzsche "Thus Spake Zarathustra" (1883-85)
I tell you:
Man must have Chaos in themselves
to be able to give birth to a dancing star.
I tell you:
You have Chaos in yourself.

• Ray Bradbury "A Sound of Thunder" (1952)
"If you bruise some kinds of plants, the effects will be piled up little by little like differential.
Here, a small error would be expanded, and it causes terrible results after 60 million years." (…)
"Such a tiny thing! A mere butterfly!"

• Isaac Asimov "Spell my Name with an S" (1959) (in "Nine Tomorrows")
"Your case is very interesting. I recommend you to change your name into Sebatinsky." (…)
"That is, you say I should change my initial? From Z to S? Is that all? (…) How will this change give effect to what?" (…)
"I don't know. It might affect you although I don't know why."

• Alfred Bester, "The Dark Side of the Earth" (1964)
"In Pleistocene, I happened to step on a small insect and killed it." (…)
"(…) I saw a phantom that the world has completely changed because of its death. "

• Gilles Deleuze & Félix Guattari "Anti-Œdipus - Capitalism and schizophrenia" (1972)
In sum, all the confrontations between absorbing force and repulsing force generate the affirmative and open sequence of various elements of intensity. These elements never represent the final state of equilibrium of a system, but represent the quasi-stable stationary state, and a subject undergoes and goes through these states one after another. (…)
Various disjunction points on the body without organs form some converging circles around the desiring-machine. In this way, the subject is produced as a remaining vestige by the side of the desiring-machine, and, as an appendage or as a part, it goes through all the states of circles and moves from circles to circles one after another.

• William Gibson & Bruce Sterling "The Difference Engine" (1990)
"They might have known this would happen, if they'd properly studied Catastrophist theory. It is a concatenation of synergistic interactions; the whole system is on the period-doubling route to Chaos!"
"What does that mean, pray?" (…)
"in layman's terms, it means that everything gets twice as bad, twice as fast, until everything falls completely apart!"

• Michael Crichton "Jurassic Park" (1991)
"The chaos theory implies two essential points. First, complex systems like weather have latent order structures. Second one tells its opposite --- even simple systems can yield complex dynamics."