"The first popular written book about this fascinating, rapidly growing discipline. It is a splendid introduction. Not only does it explain accurately and skilfully the fundamentals of chaos theory, but it sketches the theory's colourful history, with entertaining anecdotes about its pioners and provocative asides about the philosophy of science and mathematics."

The following represents my brief research notes in the reading of this work. As the theme of this website will attest, all of my research notes are available for my use on the web.

PRF Brown
Mountain Man Graphics,
Newport Beach, Australia
Southern Spring - 1997

E-Mail: prfbrown@magna.com.au

"Under normal conditions the research scientist is not an innovator but a solver of puzzles, and the puzzles upon which he concentrates are just those which he believes can be both stated and solved within the existing scientific tradition," - Kuhn.
[Page 37]

"I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives." - Tolstoy.
[Page 38]

Chaos and instability, concepts only beginning to acquire formal definitions, were not the same at all. A chaotic system could be stable if its particular brand of irregularity persisted in the face of small disturbances.
[Page 48]

The result of mathematical development should be continuously checked against one's own intuition about what constitutes reasonable biological behaviour. When such a check reveals disagreement, then the following possibilities must be considered:

a. A mistake has been made in the formal mathematical development;
b. The starting assumptions are incorrect and/or constitute a too drastic oversimplification;
c. One's own intuition about the biological filed is inadequately developed;
d. A penetrating new principle has been discovered.

Pedagogically speaking, a good share of physics and mathematics was-and is-writing differential equations on a blackboard and showing students how to solve them.
Page 67

Enrico Fermi once exclaimed,
"It does not say in the Bible
that all laws of nature are expressible linearly!"
Page 68

Chaos is ubiquituous; it is stable; it is structured. He [Yorke] also gave reason to believe that complicated systems, traditionally modeled by hard continuous differential equations, could be understood in terms of easy discrete maps.
Page 76

"The mathematical intuition so developed ill equips the student to confront the bizarre behaviour exhibited by the simplest of discrete nonlinear systems." - May
Page 80

The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a goemetry that lasted two millennia ...
Page 94

Clouds are not spheres, Mandelbrot is fond of saying, Mountains are not cones. Lightning does not travel in a straight line.
Page 94

THE KOCH SNOWFLAKE ... "A rough but vigorous model of a coastline," in Mandelbrot's words. To construct a Koch curve, begin with a triangle with sides of length 1. At the middle of each side, add a new triangle one-third the size; and so on. The length of the boundary is 3 x 4/3 x 4/3 x 4/3 ...- infinity. Yet the area remains less than the area of a circle drawn around the original triangle. Thus an infinitely long line surrounds a finite area.
Page 99

The outline of the Koch curve, with infinite length crowding into finite area, does fill space. It is more than a line, yet less than a plane. It is greater than one-dimensional, yet less than a two-dimensional form. Using techniques originated by mathematicians early in the century then all forgotten, Mandelbrot could characterize the fractional dimension precisely. For the Koch curve, the infinitely extended multiplication by four-thirds gives a dimension of 1.2618.
Page 102

"It's a single model that allows us to cope with the range of changing dimensions of the earth," he said. "It gives you mathematical and geometrical tools to describe and make predictions. Once you get over the hump, and you understand the paradigm, you can start actually measuring things and thinking about things in a new way. You see them differentially. You have a new vision. It's not the same as the old vision at all - it's much broader." - Christopher Scholz
Page 107

Trees, trees that need to capture sun and resist wind, with fractal branches and fractal leaves. And theoretical biologists began to speculate that fractal scaling was not just common but universal in morphogenesis. They argued that understanding how such patterns were encoded and processed had become a major challenge to biology.
Page 110

Gert Eilenberger, a German physicist who took up nonlinear science after specializing in superconductivity: "Why is it that the silhouette of a storm-bent leafless tree against an evening sky in winter is perceived as beautiful, but the corresponding silhouette of any multi-purpose university building is not, in spite of all efforts of the architect? The answer seems to me, even if somewhat speculative, to follow from the new insights into dynamical systems. Our feeling for beauty is inspired by the harmonious arrangement of order and disorder as it occurs in natural objects - in clouds, trees, mountain ranges, or snow crystals. The shapes of all these are dynamical processes jelled into physical forms, and particular combinations or order and disorder are typical for them."
Page 117

TURBULENCE was a problem with pedigree. The great physicists all thought about it, formally or informally. A smooth flow breaks up into whorls and eddies. Wild patterns disrupt the boundary between fluid and solid. Energy drains rapidly from large-scale motions to small. Why? The best ideas came from mathematicians; for most physicists, turbulence was too dangerous to waste time on. It seemed almost unknowable. There was story about the quantum theorist Werner Heisenberg, on his deathbed, declaring that he will have two questions for God: why relativity, and why turbulence. Heisenberg says, "I really think He may have an answer to the first question."
Page 121

Feigenbaum had discovered universality and created a theory to explain it. That was the pivot on which the new science swung. Unable to publish such an astonishing and counter-intuitive results, he spread the word in a series of lectures at a New Hampshire conference in August 1976, an international mathematics meeting at Los Alomos in September, a set of talks at Brown University in November. The discovery and the theory met surprise, disbelief, and exitement. The more a scientist had thought about nonlinearity, the more he felt the force of Feigenbaum's universality. One put it simply: "It was a very happy and shocking discovery that there were structures in nonlinear systems that are always the same if you looked at them the right way."
Page 183

A movement had begun, and the discovery of universality spurred it forward. In the summer of 1977, two physicians, Joseph Ford and Giulio Casati, organized the first conference on a science called chaos.
Page 184

"In a structured subject, it is known what is known, what is unknown, what people have already tried and doesn't lead anywhere. There you have to work on a problem which is known to be a problem. Otherwise you get lost. But a problem which is known to be a problem must be hard, otherwise it would already have been solved." - Heinz-Otto Peitgen
Page 230

The Mandelbrot Set Program ... needs just a few essential pieces. The main engine is a loop of instructions that takes its starting complex number and applies the arithmetic rule to it. For the Mandelbrot set, the rule is this:

z ==> z² + c,

Fractal Basin boundaries - [James] Yorke would rise at conferences to display pictures of fractal basin boundaries. Some pictures represented the behavior of forced pendulums that could end up in one of two final states - the forced pendulum being, as his audiences well knew, a fundamental oscillator with many guises in everyday life. "Nobody can say that I've rigged the system by choosing a pendulum," Yorke would say jovially. "This is the kind of thing you see throughout nature. But the behaviour is different from anything you see in the literature. It's fractal behaviour of a wild kind."
Page 234

Fractal Basin boundaries - Even when a dynamical system's long-term behaviour is not chaotic, chaos can appear at the boundary between one kind of steady behaviour and another. Often a dynamical system has more than one equilibrium state, like a pendulum that can come to a halt at either of two magnets pacled at its base. Each equilibrium is an attractor, and the boundary between two attractors can be complicated but smooth. Or the boundary can be complicated but not smooth. The highly fractal interspersing of white and black is a phase-space diagram of a pendulum. The system is sure to reach one of two possible steady states. For some starting conditions, the outcome is quite predictable - black is black and white is white. But near the boundary, prediction becomes impossible.
Page 235

"Nonlinear was a word that you only encountered in the back of the book. A physics student would take a maths course and the last chapter would be on nonlinear equations. You would usually skip that, and, if you didn't, all they would do is take these nonlinear equations and reduce them to linear equations, so you just get approximate solutions anyway. It was just an exercise in frustration. We had no concept of the real difference that nonlinearity makes in a model. The idea that an equation could bounce around in an apparently random way - that was pretty exciting. You would say, "Where is this random motion coming from? I don't see it in the equations.' It seemed like something for nothing, or something out of nothing." - Doyne Farmer
Page 251

"It was striking to us that if you take regular physical systems which have been analyzed to death in classical physics, but you take one little step away in parameter space, you end up with something to which all of this huge body of analysis does not apply. The phenomenon of chaos could have been discovered long, long ago. It wasn't in part because this huge body of work on the dynamics of regular motion didn't lead in that direction. But if you just look, there it is. It brought home the point that one should allow oneself to be guided by the physics, by observations, to see what kind of theoretical picture one could develop. In the long run we saw the investigation of complicated dynamics as an entry point that might lead to an understanding of really, really complicated dynamics." - Norman Packard
Page 251

"On a philosophical level, it struck me as an operational way to define free will with determinism. The system is deterministic, but you can't say what it's going to do next. At the same time, I'd always felt that the important problems out there in the world had to do with the creation of organization, in life or intelligence. But how did you study that? What biologists were doing seemd so applied and specific; chemists certainly weren't doing it; mathematicians weren't doing it at all, and it was something that physicists just didn't do. I always felt that the spontaneous emergence of self-organzatiuon ought to be part of physics. "Here was one coin with two sides. Here was order, with randomness emerging, and then one step further away was randomness with its own underlying order." - Doyne Farmer
Page 251-252

Could unpredictability itself be measured? The answer to this question lay in a Russian conception, the Lyaponov exponent.
Page 253

"The sciences do not try to explain, they hardly even try to interpret, they mainly make modesl. By a model is meant a mathematical constuct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work." - John Von Neumann
Page 273

"Dynamical things are generally conterintuitive, and the heart is no exception." - Arthur T Winfree
Page 292

Some definitions of CHAOS

The complicated, aperiodic, attracting orbits of certain (usually low-dimensional) dynamical systems.
Philip Holmes - mathematician

A rapidly expanding field of research to which mathematicians, physicists, hydrodynamicists, ecologists and many others have all made important contributions. And: A newly recognized and ubiquitous class of natural phenomena.
Hao Bai-Lin, physicist

Apparently random recurrent behaviour in a simple deterministic (clockwork-like) system.
H. Bruce Stewart, applied mathematician

The irregular, unpredictable behaviour of deterministic, nonlinear dynamical systems.
Roderick V. Jensen, theoretical physicist

Dynamics with positive, but finite, metric entropy. The translation from mathese is: behaviour that produces informatin (amplifies small uncertainties), but is not utterly unpredictable.
James Crutchfield, Santa Cruz collective

Dynamics freed at last fromt he shackles of order and predictability ... Systems liberated to randomly explore their every dynamical possibility ... Exciting variety, richness of choice, a cornucopia of opportunity.
Joseph Ford

Page 306