Dirk Bertels

he who has noble thoughts is never alone

Literature on Chaos


Here are some excerpts from the literature written by authorities on Chaos. They should help in the understanding of this great mathematical field that reaches out into the many branches of science.

Excerpts from "Chaos" - by James Gleick

ISBN 0-7474-0413-5
First published 1987


James Yorke in his "Period Three Implies Chaos" paper, proved that in any one-dimensional system, if a regular cycle of period three ever appears then the same system will also display regular cycles of every other length, as well as completely chaotic cycles.

p231 (footnote)

A 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 arithmetical rule to it. For the Mandelbroth set, the rule is this:

z -> z2 + c

where z begins at zero and c is the complex number corresponding to the point being tested. So, take 0, multiply it by itself, and add the starting number; take the result - the starting number - multiply it by itself, and add the starting number. ... The program needs to watch the running total. If the total heads off to infinity, moving further and further from the center of the plane, the original point does not belong to the set and if the running total becomes greater than 2 or smaller than -2 in either its real or imaginary part, it is surely heading off to infinity. How many times depends on the amount of magnification. For the scales accessible to a personal computer, 100 or 200 is often plenty, and 1000 is safe. The program must repeat this process for each of thousands of points on a grid, with a scale that can be adjusted for greater magnification. And the program must display its result. Points in the set can be coloured black, other points white. Or for a more vividly appealing picture, the white points can be replaced by coloured gradations. If the iteration breaks off at ten repetitions, for example, a program might plot a red dot: for forty repetitions a yellow dot, and so on. The colours reveal the contours of the terrain just outside the set proper.


To play the chaos game quickly, you need a computer with a graphics screen and a random number generator, but in principle a sheet of paper and a coin work just as well. You choose a starting point somewhere on the paper. You invent two rules, a heads rule and a tails rule...Say, move 2 inches to the north-east, or move 25% closer to the center. Now you start flipping the coin and marking the points accordingly. If you throw away the first 50 points, you will find the chaos game producing not a random field of dots but a shape, revealed with greater and greater sharpness as the game goes on. ...The act of writing down a set of rules to be iterated randomly captured certain global information about a shape, and the iteration of the rules regurgitated the information without regard to scale.

Barnsley quickly found that he could generate all the now classic fractals from Mandelbrot's book. Mandelbrot's technique had been an infinite succession of construction and refinement...By using the chaos game instead, Barnsley made pictures that began as fuzzy parodies and grew progressively sharper. No refinement process was necessary: just a single set of rules that somehow embodied the final shape.

Barnsley's central insight was this: Julia sets and other fractal shapes, though properly viewed as the outcome of a deterministic process, had a second, equally valid existence as the limit of a random process.


To that extent the role of chance is an illusion, "Randomness is a red herring," Barnsley said. "[Randomness] is central to obtaining images of a certain invariant measure that live upon the fractal object. But the object itself does not depend on the randomness... It's giving deep information, probing fractal objects with a random algorithm. Just as, when we go into a new room, our eyes dance around it in some order which we might as well take to be random...The object exists regardless of what I happen to do."


Farmer said, "On a philosophical level, [chaos theory provided an] operational way to define free will, in a way that allowed you to reconcile free will with determinism."

Jack Cohen and Ian Stewart - The collapse of chaos

ISBN 0-14-029125-3
First published 1987


Attractors are the things that the dynamics converge toward if you wait long enough; but once they reach the attractors, they promptly diverge again - and drastically... Anything off a chaotic attractor is "folded" towards it; but anything on it is "stretched" in an unpredictable way - except that one thing is predictable: It always stays on the attractor...

... The butterfly effect means that nearby points on the attractor tend to separate - but they stay on the attractor. Think of a ping pong ball in an ocean, with complicated currents at the surface. If you release the ball from below the surface, it floats upward. If you drop it from above, it falls downward. It is attracted to the surface... The ocean surface is the attractor, and the ball always ends up there; but the dynamics on the surface can be very complex and unpredictable.


Complexity at any given level is a consequence of the operation of relatively simple rules one level lower down. Simplicity breeds complexity through sheer multiplication of possibilities; organisation derives from the tidy simplicity of the laws.


One of the great surprises of chaos theory is the discovery of totally new simplicities, deep universal patterns concealed within the erratic behaviour of chaotic dynamical systems. The first of these unexpected simplicities was discovered by the physicist Mitchell Feigenbaum, and is known as the Feigenbaum number. We'll explain it in a physical interpretation: If a small quantity of liquid helium, cooled very close to absolute zero, is heated from below, it forms tiny convection cells, in which the helium circulates and carries heat upward. If the temperature at the bottom is increased a little, then the cells begin to wobble periodically:

wobble - wobble - wobble - wobble ...

At a higher temperature, pairs of consecutive wobbles become slightly different:

wobble - wobble - wobble - wobble - wobble - wobble - wobble

The period doubles in length; you now have to wait for two wobbles before everything repeats...As the temperature rises, the wobbles group into fours...etc... This doubling of the period by the creation of ever finer differences between consecutive sequences of wobbles is called a period doubling cascade. Each successive step in the cascade occurs as the result of an ever-smaller rise in the temperature.

In mathematical models there is a particular critical temperature, and when this is reached, the period has doubled infinitely often, resulting in chaos. The period-doubling cascade is a route from order to chaos. It's important because it is one of the commonest such routes.

...Feigenbaum discovered that (taking our helium example) the amount by which the temperature must rise, in order to double the period, decreases geometrically as the period gets longer. Each such increment is roughly 1/4.669 times as long as the previous one.

...A decreasing geometric series is an easy way to cram infinitely many events into a finite space, and any such series must have some common ratio. But when Feigenbaum tried different mathematical equations with a period doubling cascade, he got the same ratio, 1/4669...a new, totally unexpected simplicity, emerging from some of the most complex behaviours known to mathematicians.


Colour is an emergent phenomenon; it only makes sense for bulk matter...Emergence is the source of new simplicities...it helps make respectable the idea that a collection of interacting components can "spontaneously" develop collective properties that seem not to be implicit in any way in the individual pieces.

Emergent simplicities collapse chaos; they bring order to a system that appears to be wallowing hopelessly in a sea of random fluctuation.

...For example, your continued existence depends upon the atmosphere maintaining its normal range of pressures and its normal oxygen content. Because you are very big, on atomic scales, and your movements are very slow, your reaction to gas molecules is insensitive to tiny details. You "smooth out" the fluctuations by not noticing them, so you can build reliable structures upon the statistical features. Your lungs trawl the atmosphere for schools of oxygen molecules, and it doesn't matter how they get into the net as long as the total catch is its usual size. A very tiny, fast moving creature would have to hunt down individual gas molecules like a hunter-gatherer on the Savannahs, and would be terrible vulnerable to random shortages.

...Statistical regularities are certainly one important and widespread mechanism for emergence...However many emergent features do not come from statistics. There is nothing statistical about PI, the Feigenbaum number, the Mandelbroth set - or chlorophyl., DNA or homeotic genes, for that matter.


Simplexity is the emergence of large-scale simplicities as direct consequences of rules ... The real-world parallel to simplexity is exploited to good effect by nature. Mars is indeed composed of atoms; these do - as far as we can tell - obey Newtonian dynamics to a good degree of approximation, and Mars conveniently moves in a near ellipse. This can't be coincidence. There must be a kind of natural simplexity in operation as well, otherwise Mars wouldn't behave in such a simple manner. The natural simplexity should work for the same kind of reason that the theoretical one does: The net effect of all those interatomic forces, on another collection of atoms as distant as the sun, causes Mars to move just as it would if its mass were all concentrated at its centre. This is no more than a restatement of the previous reductionist rhetoric, but one that makes the assumptions explicit. The features in the theoretical domain are congruent to the instances in the natural one.

... Simplexity merely explores a fixed space of the possible. Complexity enlarges it. And both processes collapse the underlying chaos, producing stable features from a sea of complexity and randomness.


Both the elephant and the giraffe have a common feature: "Long thing for drinking water without kneeling down and tearing off leaves without hopping into the air."...in terms of external constraints such as competition for food and water, the explanations of the evolution of trunks and necks probably parallel each other closely...

... Complicity arises when simple systems interact in a way that changes both and erases their dependence on initial conditions. The hallmark of complicity is the occurrence of the same feature or features in systems whose rules are either known to be very different, or are expected to be very different if only we could find out what they are. This carries an important consequence: Complicity is a convergent process; it homes in on the features regardless of fine detail in the rules...The prime example of complicity is evolution, for which the two systems of rules are the chemistry of DNA and the systematic ways in which organisms interact with their environment. Consider the animal feature "wings". Such a feature has evolved several times, starting from entirely different circumstances, in insects, pterodactyls, bats, birds...All 4 evolved wings because they solved the identical problem in their environment: to get off the ground. They built a bridge to "Flight island" and expanded the space of the possible.


Our brains have evolved an impressive ability to detect features. Features are generated in nature by the collapse of chaos, and they provide a quick-and-dirty method for anticipating events in our environment so that we can respond more rapidly to possible threats...But feature detectors are themselves features, so a generalised feature detector will be self-referential. We give the label "consciousness" to our feature-detection system. We become conscious of a feature of the world when our brain detects it. Therefore consciousness is self-referential, that is, we are conscious that we possess consciousness. We tend to think that this property of consciousness is the most surprising one, but actually, it is a simple consequence of the generality of our feature-detecting system...What needs explaining is not the self-referential nature of consciousness, but how it can take a huge quantity of partially structured sensory data and extract important features - the same features that "natural" interactions see.

... It is complicity, not quantum mechanics, that leads to consciousness. We can see why consciousness evolved: A good way to predators or find mates or generally manage your life is to decide between alternatives rather than blindly following predictable rules.