Nonlinear Mathematics

Alan Mayne

Nonlinear mathematics has advanced explosively in recent decades. Many of its more recent developments have become well-known to the public, partly because of their very broad practical significance, and partly because some of them produce highly artistic coloured images. The realisation, that nonlinearity occurs very extensively in the universe, is totally transforming our scientific and thus our philosophical world views. It is not very easy to define 'nonlinearity' in a way that can be understood by non-mathematicians. A 'linear system' can be defined very roughly as one whose whole is the sum of its parts. More generally, in a 'nonlinear system', the whole is no longer equal to the sum of its parts. Linear systems are generally very much easier to handle mathematically than nonlinear systems, and practical solutions of most nonlinear mathematical problems became feasible only with the advent of modern computers. From about the 1960s on, nonlinear mathematics began to branch out from nonlinear dynamics into new sub-branches such as chaos theory, complexity theory, fractals, and catastrophe. All these aspects are covered in the Key Texts below, some of which are popular books for the general reader; the others are intended for mathematicians but include aesthetic patterns of wider appeal. Mathematical expositions of nonlinear systems and nonlinear dynamics are omitted.

Chaos theory is the first branch of nonlinear mathematics to become widely known to the public, and provides mathematical descriptions of 'chaotic' phenomena themselves and of transitions from order to chaos or from chaos to order. Areas 'at the edge of chaos', that is, at the transitional region between ordered and chaotic phenomena, are often of special significance. Some types of chaotic phenomena, for example in weather, are extremely sensitive to their initial conditions, so that their results can be totally unpredictable in practice! Complexity theory enables scientists in many different disciplines to abandon the linear, reductionist thinking, that has dominated so much of modern science, and explore new concepts about interconnectedness, coevolution, nonlinearity, structure, order, and chaos. This multidisciplinary science is leading towards an entirely new way of thinking, a new paradigm, about nature, life, human, social, and economic behaviour, and the universe itself. Thus it is a most promising breakthrough in unified scientific thinking, with many practical applications. Fractal mathematics describes broken-up shapes that are irregular and have the same degree of irregularity at all scales. A fractal object is 'self-similar' in that it looks similar when examined from nearby or at a distance. Many fractal shapes occur in nature, for example, coastlines, mountains, clouds, and the forms of many plants; the distribution of galaxies in the universe is fractal. Catastrophe theory was created by René Thom, who was a pure mathematician but also a 'natural philosopher', who thought deeply about the natural order and how it is reflected in scientific theories. It is qualitative, not quantitative, but well adapted to describe and even predict the 'shape' of certain processes, and provide insights complementary to those obtained from quantitative mathematical analyses and calculations.

Alan Mayne is a self-employed author, editor, personal computer specialist and researcher. Latest books: 'Into the 21st Century: A Handbook for a Sustainable Future' (co-author), 'Resources for the Future: An International Annotated Bibliography for the 21st Century' . editor of 'New Paradigms Newsletter'.

Key texts

Gleick, J. (New Ed. 1998) Chaos: Making a New Science. London: Minerva.
The best-known popular exposition of chaos theory, and an excellent survey of for those without mathematical knowledge. It gives a fascinating account of its evolution as a new branch of nonlinear mathematics during the 1960s and 1970s, and outlines examples of its many applications to science and technology and to practical applications like weather forecasting and stock market prediction.

Hall, N. (Ed.) (1992) The New Scientist Guide to Chaos. London: Penguin Books, & New York: Penguin Books USA. ISBN 0-14-014571-0.
Another good popular exposition of chaos theory, which gives many examples of its scientific and other applications. It also gives some examples of fractal shapes and patterns.

Lorenz, E. (2nd. Ed. 1995) The Essence of Chaos. London: UCL Press. ISBN 1-85728-187-x.
A fascinating general introduction to chaos theory by one of the mathematicians who pioneered its development. Especially as it has many illustrations of beautiful mathematical curves and patterns, it is mostly accessible to non-mathematicians. It explains the basic phenomena of chaos, and describes various natural phenomena where it arises. It concentrates most attention on chaotic phenomena which occur in weather.

Waldrop, M. Mitchell (2nd. Ed. 1993) Complexity: the Emerging Science at the Edge of Chaos. London: Viking. ISBN 0-670-85045-4. (1st. ed. 1992) New York: Simon & Schuster.
Lewin, R. (1993) Complexity: Life at the Edge of Chaos. London: Dent. ISBN 0-460-86092-5.
These two books, like Gleick's book, discuss, from complementary viewpoints, the work of the remarkable Santa Fe Institute for the study of the sciences of complexity, in New Mexico, USA. By introducing the personal stories of the lives and thinking of those most directly involved, and vividly conveying the excitement of their work, they bring out the essence of the new ideas that they discuss. They have useful lists of further reading for those who wish to explore the relevant mathematical and scientific ideas further.

Casti, J. L. (1994) Complexification: Explaining a Paradoxical World Through the Science of Surprise. London: Abacus. ISBN 0-349-10612-6.
The author draws on his experience of working at the Santa Fe Institute to show how complexity theory can become the basis for a science of surprise. It shows how the theory challenges and gives counter-examples to each of several 'conventional intuitions'.

Cohen, J. and Stewart, I. (1994) The Collapse of Chaos: Discovering Simplicity in a Complex World. London & New York: Viking. ISBN 0-670-84983-9.
Adopts a complementary approach to Waldrop and Lewin, by inquiring how simplicity in nature is generated from chaos and complexity. Asks why simplicities exist and can persist in a complex universe, and derives simplicity from the interaction between chaos and complexity. Finds that many different complex systems contain the same simple large-scale patterns, because such patterns do not depend on detailed substructure.

Coveney, P. and Highfield, R. (2nd. Ed. 1996) Frontiers of Complexity: The Search for Order in a Chaotic World. London: Faber and Faber. ISBN 0-571-17922-3. (1st Ed. 1995) New York: Ballantyne Books.
The most comprehensive survey of complexity theory for the intelligent general reader, covering a wide range of different aspects, including self-organisation, artificial life, neural networks, robotics, evolutionary game theory, and computer genetics. Provides a good guide to the future possibilities opened up by complexity theory.

Bak, P. (1996) How Nature Works: The Science of Self-Organized Criticality. New York: Springer, New York, & Oxford: Oxford University Press. ISBN 0-387-94791-4.
Introduces a concept of 'self-organised criticality' to explain various complex patterns that repeatedly occur in nature and in human affairs, and gives examples from various branches of the natural sciences and especially from economics. Like many other systems, economies show various forms of instability and are liable to fluctuations of activity that can occasionally be very large.

Anderla, G., Dunning, A. & Forge, F. (1997) CHAOTICS: An Agenda for Business and Society in the 21st Century. London: Adamatine Press. ISBN 0-7449-0137-5.
Applies the perspective provided by what the authors call 'chaotics', a combined mathematical theory of chaos and complexity, to develop what they view as a truer understanding of the processes of the real world. It suggests various ways in which chaotics could be applied to help improve the human situation.

Mandelbrot, B. (2nd. ed. 1983) The Fractal Geometry of Nature. New York: W. H. Freeman. ISBN 0-7167-1168-9.
Book by the originator of fractal mathematics, presenting the basic mathematical theory. The mathematician but also the general reader will appreciate its many illustrations of fractal shapes and patterns, some in colour and of great beauty.

Woodcock, A. and Davis, M. (1978) Catastrophe Theory. New York: E. P. Dutton. ISBN 0-525-07812-6.
A good preliminary explanation of catastrophe theory and some of its applications for non-mathematicians. It outlines the seven 'elementary catastrophes', each indicating how a particular type of sudden qualitative change of system behaviour can occur. It gives examples of how catastrophe theory can throw light on problems in physics, chemistry, biology, animal psychology, human psychology, sociology, politics, and economics.