Chaos Theory: Introduction

The rules, by which the elements of nature interact, are simple. Yet the res­ult, when seen on a large scale, appears chaotic. But it is not. It is merely complex. It is also benign. So perhaps, if every individual human relation­ship were governed by similar rules then a benign egalitarian society should be possible. [Português]

Science and engineering — and perhaps many other disciplines — are replete with misnomers. This is because their names are given at a very early stage when the phenomenon being named isn't as yet very well understood. In the context of Chaos Theory, the word 'Chaos' is definitely a misnomer. Chaos means "complete disorder and confusion". Although the phenomena that Chaos Theory attempts to describe appear as swirling masses of complete disorder and confusion, they are not. On the contrary, the motions [or dynamics] of these phenomena are governed — as is everything else — by the simple deterministic laws of physics. It is simply that this deterministic motion is extremely complex. Hence, what is colloquially still known as Chaos Theory was renamed as the study of "Complex Dynamics".

Formative Interest

Even from my A-level days at school, I loved maths. But this affection always suff­ered an undertow. This was because maths, as I knew it, only ever seemed to deal with what I saw as special cases. It could find the area of a square, a circle and even areas beneath regular curves. But all these are only a few special cases of the irregular shapes that make up the real world. Later I felt disappointed that so-called 'solvable' differential equations only related to a few very special cases of motion like that of a pendulum or electronic oscillator. They could not deal with the unlim­ited variety of motion we see every day. Chaos Theory was thus a happy revelation.

Lorenz's strange attractor showed that even the nastiest of differential equations had what we call 'solutions'. We previously didn't have the eyes to see them. I thus became intrigued by the effect of iterating simple difference equations. Conse­qu­ently, along with probably thousands of others, I could not resist writing programs to explore these effects for various such equations. This in turn led me to experi­ment with further programs to display these equations' bifurcation maps. And of course I could not miss out on writing generators for the infamous Mandelbrot Set and Hénon's Strange Attractor.

Nature of Systems

A system is a group of elementary functional units [active components] that are somehow interconnected to work together as parts of a larger and more complex functional unit. The components of a system each has its different roll to play with­in the whole. The function of the system, however, is in general much more than the sum of its component functions. As they say, "a system is more than the sum of its parts". This is because a system's overall function is derived not merely from the functions performed by its components, but much more from the manner — the plan, pattern or form — in which those component functions are arranged or put together. So a system's overall function is something that is way beyond what is merely inherent to the total functionality of its components.

A computer is a system. It comprises elemental functional units — a CPU [central processing unit], a memory, input devices, storage devices and output devices. Each elemental unit performs a specific and different function. So when they are all interconnected electrically to work together, they form a composite that we call a computer system. The human body — with its brain, heart, lungs, kidneys, liver and so on — can equally well be described as a system. So systems can be natural as well as artificial.

A computer and the human body belong to a class of system known as machines. They have components that are arranged in what are ostensibly rigid spatial rela­tionships with each other, each component having its own dedicated and specific function to per­form within the overall functionality of the machine as a whole.

The kind of phenomena that Chaos Theory attempts to describe may be rightly call­ed 'systems'. But they are not machines. Their components are not necessarily held in fixed spatial relationships with each other. In other words, such systems are not rigid machines but have more the nature of a fluid. Rather than having separate and different elemental rolls to play within a composite endeavour, their compon­ents are generally identical in size, structure and function. And rather than being fix-wired to each other like the components of a computer, they communicate with each other during temporary encounters using a simple universal protocol. This kind of system is referred to as a complex-dynamical system.

A complex dynamical system gains its overall character not from its components, but from the relationships between them. It can have global characteristics that are not present in its individual parts because these global characteristics are chara­cteristics of the relationships between the individuals it comprises and not chara­cteristics of the individuals themselves. Notwithstanding, the characteristics of this fundamentally different kind of system are generally less complex than their comp­onents because they are "more than the sum total" of the relationships between individuals: not "more than the sum total" of the individuals themselves.

Examples of complex dynamical systems are the Earth's weather, animal & insect communities and human societies & economies. Differently from the others, how­ever, the latter two are not left free to self-regulate solely according to the laws of physics, which include the protocols by which their basic components interact. Gov­ernments, which still seem to view a socio-economy as some kind of Keynesian machine, superimpose mechanistic rules upon a complex dynamical system, with hierarchies of enforcement to constrain its components (its citizens) to conform.

But hierarchies have no part in complex-dynamics. The imposition of hierarchical con­trol can, at best, attenuate or distort the proper workings of a complex dyn­am­ical system. At worst — once all the 'legal loopholes' have been closed — it can drive a socio-economy into self-destructive catastrophe. The saving grace so far has been the enormous natural resilience of complex dynamical systems. But the signs are that this resilience is nearing its limit.

Practical Lessons

Will technology and mass-media marketing eventually drive the global economy into chaos? After all, almost every national economy even now exhibits boom/bust peri­odicity of increasing violence and erraticness. Bearing in mind that this behav­iour is rooted in no­thing more than a fundamental property of numbers, what can the politicians — or any­body else for that matter — do to stop it reaching the point of self-destruction? Well, they could stop trying to control it.

One could wind technology back 100 years. But it would be unsafe to do that uni­laterally. I think a better way may be to study chaos to find out how a techno­log­ically advancing tightly-coupled global economy could be made to follow a be­nign strange attractor. This would be one within the bounds of which no individual could ever suffer violence, humiliation, deprivation, or poverty.

The complex behaviour of the world's weather is determined by the simple rules of engagement which exist between the atoms and molecules of the atmosphere, like zillions of little finite-state machines continually exchanging messages. The com­plex behaviour of society is determined by the protocols of social and economic ex­change, which exist between all the individuals that make up society. If the fractal protocols that govern these individual relationships were properly engineered, and followed, then the whole world could be a better place. However, doing this would require poli­ticians to change their view of what they govern. They would have to replace their absolute holistic views of 'world', 'nation' and 'corporation', with a relativistic fractal view in which every individual on this planet is wholly included and seen as equally precious.

© March 1999 Robert John Morton | HOME | NEXT