It is difficult for professional scientists, much less the general public, to distinguish excessive hype from solid scientific achievement. Chaos has given us both.

John Franksfootnote1

There is currently a great deal of excitement about the notion of chaos. Science writer James Gleick’s book on the subject has become a bestseller,footnote2 there have been numerous articles in the general press and programmes on television, and New Scientist ran an extensive series of weekly articles on the relevance of chaos theory to a range of disciplines.footnote3 Terms from chaos theory, such as ‘butterfly effect’ and ‘strange attractor’, have entered the common vocabulary, though not always used very precisely. While not without precedent, this is an unusual situation for a subject that is concerned first and foremost with developments in mathematics, some of which are, in fact, of fairly mature vintage. Moreover, the connection between the mathematical formalism of chaos theory and events in the real world is considerably more problematic than is often implied in general expositions. Gleick’s book, for example, asserts such connections far too glibly, glossing over the complex relations between theory, model and real system. His claim that chaos theory is the new paradigm for science should, at least at this stage, be viewed with considerable caution. In his treatment, the borderline between mathematics and real systems becomes blurred, to say the least, while the mathematical content of the new developments remains somewhat obscure. What Gleick presents is a tale of discovery, emphasizing individual achievements, and on that level his book is a very good read. A mathematically more serious treatment of the subject is presented by Ian Stewart in his study Does God Play Dice?footnote4

The new notion of chaos is only one of the results to have emerged from recent developments in nonlinear dynamical systems theory. The basic idea is quite simple: that a good proportion of simple deterministic systems are, at least in theory, capable of producing extremely complex and to all intents and purposes unpredictable behaviour. Chaos in this context is deterministic disorder. There is, moreover, a great deal of evidence that similar forms of behaviour occur in a wide variety of real systems. The surprise has been the realization that complex and irregular behaviour in real systems need not reflect a parallel complexity in the nature of the system itself.

This article aims to present an intuitive picture of how simple systems can give rise to chaotic behaviour, by discussing briefly some of the new developments in nonlinear dynamical systems theory, including their background and potential implications. To keep ideas clear and non-contentious, the discussion will be confined primarily to the exact sciences. In principle, however, these developments could have an important bearing on the many fields where the origin of complex behaviour is under study—especially where notions of what is scientific are informed by developments in physics.

It is now thought that examples of nonlinear dynamical systems are all around us: waterfalls, biological organs, ecosystems, the biosphere as a whole, socio-economic systems, the Earth’s climatic system, the solar system, and so on. Nonlinear dynamical systems theory is concerned with the construction of mathematical models of such systems and with the investigation of the properties of the models. These properties are taken to provide insights into the behaviour of the real systems represented by the models. In order to keep clear the distinction between real system and mathematical model, it is necessary to examine the definition of certain terms used in nonlinear dynamical systems theory. Furthermore, a focus on the restrictive assumptions underlying these definitions should highlight the limitations in the domain of relevance of the new developments.

In general, the term system is taken to mean anything that has parts and can at the same time be treated as a single whole. In dynamical systems theory, however, the term is also used in a more restricted sense, referring to the set (system) of mathematical equations used to describe the (real) system. In other words, the mathematical model itself is generally called a (dynamical) system. This point deserves to be stressed, as it is a source of much confusion.

A system is said to be dynamical if its state (or mode of behaviour) changes with time. The state of a system can be represented by a number of quantities which, in a dynamical system, vary with time. The minimum number of such quantities (called variables) necessary to characterize the system is called the dimension of the system. Consider, for example, an isolated habitat where populations of three different species of organism interact. The dynamical system consisting of the three populations (assuming it is closed) would be represented by a set of exactly three equations. Each of these equations would relate the rate of change in one of the variable quantities (that is, the level of one of the populations) to some combination of the other quantities. In this example, three interdependent variables are involved, so the dimension of the system is three. In general, systems represented in terms of n interdependent variables are said to have dimension n, where n is any number.