Keywords: Chaos, Catastrophe Theory, Non-Linear Systems, Chaos in Health

Citation:
Jones, P. (2000) An Introduction to Chaos,
<>, Accessed

Introduction

The still relatively new science of chaos has stimulated interest in many diverse fields, for example: meteorology, economics, physiology, molecular physics, and astronomy. This page provides a brief introduction to how chaos and the world of the non-linear came to the attention of so many disparate disciplines. It also outlines the concept in basic terms - below, with examples and links to resources.

What is chaos?

The idea behind 'chaos' is that order underlies what appear to be chaotic systems. The dependence of events and phenomena upon initial conditions has long been recognised:

"For want of a nail, the shoe was lost;
	For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
	For want of a rider, the battle was lost;
For want of a battle, the kingdom was lost!"

Technically this is known as sensitive dependence on initial conditions. A small change in the initial input into a system can have pronounced effects on the outcome. Over time the outcome might diverge greatly from that which might be expected. Chaos lies in the realm of the non-linear. Non-linear phenomena are usually not very accessible to investigation. That changed with the arrival of computers, and taking advantage of tasks that computers do best:

Gleik's popular book CHAOS: Making A New Science acknowledged the work of Lorenz, who first recognised sensitive dependence in attempts to model the weather. The computer proved the key. Computers opened a door, knocked upon years before by Poincarè and Verhulst, but they did not have the tools. The secrets were out of reach, until the experiments and discoveries of Lorenz and Benoit Mandelbrot.

Chaos centres on the use of iterative processes, in which a product from a calculation is used as input repeatedly, to form a feed-back process. Computers were 'born' to do such calculations, for example at: http://mathforum.org/alejandre/applet.mandlebrot.html (using a java applet), as the above table seeks to highlight. The results of number crunching were shown in two notable 'coffee table' books by Peitgen et al. (1986) - The Beauty of Fractals and The Science Of Fractal Images.

How are these produced? The short answer lies in some quite simplistic looking math.

The iterative formula in The Beauty Of Fractals is:

The details can be found in the literature, which points out that the parameter c is selected before the iteration begins.

The results displayed in graphical form are truly astonishing. Video examples of fractals with 2D and 3D vistas are even more stunning. The images tantalise and fire the imagination of many people who, like myself, do not have a natural aptitude for mathematics. Today fractal images are rather passé. They still adorn T-shirts, the club scene, poster art and literature, as in Arthur C. Clarke's (1992) SF novel The Ghost from the Grand Banks.

The 1980s saw chaos feature in many science programs on television, radio and in learned journals. Home microcomputers gave access to worlds other than those inhabited by space invaders. The scale and complexity of the Mandelbrot Set - an icon of late twentieth century science - is stunning, awe inspiring and incomprehensible:

'.. if the Mandelbrot Set was drawn so that it was 100 Kiloparsecs across (a suggested radius of our Galaxy), then a magnification of 3.5 x 10(to the power 31) would represent one hydrogen atom! Interestingly, at these extreme magnifications, one still sees miniature M – sets and the same sort of patterning around them. Editorial, (Fractal Report, 1991)

As mention of Poincarè attests, these developments did not happen overnight. Important ideas lie behind such terms as:

A Catastrophic History

Mathematically derived theories exist which have been applied in social modelling. One example is catastrophe theory; a theory based on mathematical principles, first described by Rene Thom (1975) and used to conduct research into discontinuous phenomena. Zeeman applied catastrophe theory in mechanics, economics, ecology, and biology, in an attempt to derive new models. Social models were also investigated. For example, Zeeman and several psychologists studied the discontinuity represented in the occurrence of prison riots; namely, the sudden shift from a stable (social) state to an unstable state. The goal being that the nature of discontinuity (or 'catastrophe') arising in a riot might be given to analysis, prediction and hence prevention. These initial studies in the 1970s met with varying degrees of success, such that general application of catastrophe theory provoked debate Rosenhead (1976) in New Scientist, and Woodcock & Davis (1980).

Ethical considerations aside, one of the problems immediately apparent is the data. In the prison study the variables analysed included - the levels of tension and alienation among the prisoners. This was done retrospectively (in 1975) on data from a riot that occurred in 1972. This extreme example highlights the many problems faced. In such an information rich environment which data is relevant? What aspects are significant?

Poston & Stewart (1978) consider the applications of catastrophe theory and in so doing note:

'Thom's prime motivation for studying catastrophes (in the broad sense) was the search for a framework of theory to bind together the enormous quantity of observational data available in biology. It is perhaps ironic that little of the extant data can be used to test the theories that have emerged, because the phenomena they predict often require new kinds of data for experimental confirmation! Thom's theories are very broad and philosophical, giving the style of modelling but not the details.'

Although many disease processes are insidious in onset, the act of becoming 'sick' can prove to be a catastrophic event. Heart disease is a good example, with fatalities from myocardial infarction occurring in younger age groups. Using heart disease as an example of a 'physical catastrophe', we can also recognise the accompanying 'social catastrophe' resulting from this and other pathology, i.e., suicide. Woodcock & Davis show how catastrophe theory may be used to model the effects of predisposing and precipitating causes in specific medical conditions, including schizophrenia. Catastrophe theory can show graphically as in the figure, and in mathematical form the dynamic nature of the smooth and sudden changes which occur in a variety of systems. At the time of publication of Thom's paper on catastrophe theory, many claimed a revolution was about to occur. This new theory being the tool and key to new methods and knowledge: progress.

The stimulus in social research provided by catastrophe theory per se did not last long. As soon as 1978 Poston and Stewart remarked that:

'Catastrophe theory is already beginning to disappear. That is, 'catastrophe theory' as a cohesive body of knowledge with a mutually acquainted group of experts working on its problems, is slipping into the past, as its techniques become more firmly embedded in the consciousness of the scientific community.' Ibid. p.426

What is new quickly becomes commonplace. A 'current' scientific icon soon becomes dated with the other scientific paraphernalia of the day. Catastrophe theory still has a place, spanning science and art, informing dynamical systems research and extension of literary forms (Hasegawa; Vierling).

Within Woodcock and Davis' book there is mention of 'chaotic systems' and 'strange attractors'. It is surprising these terms are not mentioned more frequently, as soon after these terms were applied, catastrophe theory was engulfed by a much larger fish. This 'larger fish' has been described by Joseph Ford as:

'Dynamics freed at last from the shackles of order and predictability. Systems liberated to randomly explore their every dynamical possibility .... Exciting variety, richness of choice, a cornucopia of opportunity.' Gleik (1989) p.306

Crutchfield in Gleik (1989):

'Dynamics with positive, but finite, metric entropy. The translation from mathese is: behaviour that produces information (amplifies small uncertainties), but is not utterly unpredictable.' Ibid.

Entry into Chaos

CHAOTIC AFFAIRS OF THE HEART

For a long time disease processes have been described in dynamic terms; time and change are key aspects in the pathology, overt characteristics of diseases. Mathematical pathology - chaos - has been transposed into medical terminology. With disease characterised as mathematical health, being viewed as the predictability and differentiability of this kind of structure Gleik (1989); May (1989).

From this perspective death is viewed as a state of equilibrium. The illustrations of the circulatory system first rendered centuries ago, highlight just one of the fractal structures within our bodies. Heart arrythmias and blood clotting have been associated with chaotic behaviour. SDPs Symmetrized dot-patterns are a modern adjunct to these realisations, as illustrated in Pickover's book, applied to cardiology:

'The symmetrized dot-pattern ... can be used to represent normal and pathological heart sound (mild mitral stenosis, and mitral regurgitation) ... Unlike the ECG which measures electrical activity of the heart, the SDP described here uses acoustic input. The symmetrized dot-pattern (SDP) characterises waveforms using patterns of dots and requires very limited computational time as prerequisite.'

Pickover also mentions the representation of DNA by DNA vectorgrams; these 'can be used to search for patterns in the sequence of bases in DNA.' Ibid. Applications in data and information processing are now well established, in speech recognition, character recognition, data compression for communications and graphics manipulation. The chaos we see in the external world is to be found in our grey matter. One article describes and illustrates the chaotic behaviour involved in the perception of smell. Freeman (1991) New perceptions of the familiar, made accessible by the computer. Gleick cites Kuhn:

"It is rather as if the professional community had been suddenly transported to another planet where familiar objects are seen in a different light and are joined by unfamiliar ones as well." p.39

CHAOS: A CASE OF HEART RULING THE HEAD?

Since mathematics and logic form the basis of the most indubitable form of knowledge - physical laws and analytic truths - we might expect that mathematically derived theories would be successful, when applied in the everyday world. Unfortunately, the application of mathematically based theories in the social sciences does not guarantee the production of 'scientific' results. As Harre (1985) informs us: 'We must beware of supposing that the principles of correct reasoning, say in mathematics, hold good for other subject matters, say chemistry. This would be like supposing that all languages have the same grammar.' Social models are mathematical isomorphisms and whilst their abstractness makes them very versatile, care must be taken in the way they are applied. The future, however, should be very exciting as we enter what will surely be a golden age of modelling. Concerning chaos and mathematics Vivaldi (1989) writes:

'I cannot help wondering what mathematics would be like if the human could perform 1012 arithmetical operations per second. It would certainly be very different, and so would be our mathematical descriptions of the physical world. Machines of that speed are now being conceived. Mathematics will change.'

Perhaps in addition the social sciences are set to change? Are already changing in fact? So, if the scientific method is said to comprise theoretical science and experimental science, then there is now a third, that of computational science.

I think the next century (21st) will be the century of complexity.
Stephen Hawking

CHAOS: SOCIAL AFFAIRS

The social sciences have not been slow to adopt developments on other areas of thought. Cybernetics with its attendant systems theory continues to influence therapeutic intervention both in theory and practice, as definitions for terms Poiesis and Autopoiesis by Varela, et al. (1974) show below:

This mode of description sees living beings as systems that produce themselves in a ceaseless way. An autopoietic system is at the same time the producer and the product.

Perhaps this is a sign of an almost natural progression towards consilience integration of knowledge sources (the universe getting to know itself?), an effort to assume control over social data via representation and manipulation of the fuzzy world of human affairs. An unconscious collective goal of psychology, sociology, anthropology, economics and other disciplines. Researchers have realised that the distinction between the 'pure' sciences and the social sciences is less clear than originally thought.

As per the unsystematic review of current opinion suggests that expectations may have been raised in terms of the delivery of ready-to-deploy applications. There is a proliferation of papers claiming to apply the 'new' mathematics including non-linear dynamics to the social sciences. Lipscomb (2003) challenges these claims, finding that many depend upon a lack of mathematical awareness in the readership. Lest the horse suddenly leave the temptation is always present to jump on-board, without awareness of the direction or exact purpose for the journey. In nursing similar problems arise (Marks-Maran, 1999; Rolf, 1999). This should not deter exploration. The lack of clarity may also be a consequence of the maturity of thought in a given field. The struggle between the soft and hard sciences may be a sign of immaturity. Our understanding of information and use of the concept may be at a similar point to that of energy in the 18th century (von Baeyer, 2003).

In the introduction to visualization a table of references demonstrated the importance of didagrams and related visual tools in health care. In discussing chaos, the exercise is repeated, but with the intention to show how chaos and the non-linear realm can conjoin h2cm and all the themes within this website.

Nursing has nonetheless, been at the forefront of efforts to apply non-linear dynamics to health and social care, describing strengths and issues arising (Chinnis, et al. 1999; Haigh 2002). Academic centers have arisen devoted to study this topic (Center for Nonlinear Science, College of Nursing, Texas Woman's University). This fusion must also pay dividends for tools such as Hodges' Health Career Model, by creating demand for a new generation of blackboards.

top : home

© Peter Jones 2000

References

Chinnis, A., White, K.R. (1999) Challenging the dominant logic of emergency departments: Guidelines from chaos theory, The Journal of Emergency Medicine, 17: 6, 1049–1054.

Clarke, A.C. (1992) The Ghost from the Grand Banks, Orbit.

Coppa, D.F. (1993) Chaos theory suggests a new paradigm for nursing science, J. Adv. Nurs., 18: 6, 985-991.

Daintith, J., Nelson, R.D. (Eds.) (1989) Dictionary of Mathematics, Penguin.

Danette, P. (2004) Spreading Chaos: The Role of Popularizations in the Diffusion of Scientific Ideas, Written Communication, 21, 1, 32-68.

Editorial (1991) Fractal Report, Reeves Telecommuncations Laboratories Ltd, Truro, UK., 17, 18.

Freeman, W. (1991) The Physiology of Perception, Scientific American, SA Inc, 264: 2, 34-41.

Gleik, J. (1989) CHAOS: Making A New Science, London, Cardinal, 306.

Ibid. 39, 298

Gordon K. (2003) The Impermanence of Being: Toward a Psychology of Uncertainty, Journal of Humanistic Psychology, 43, 2, 96-117.

Haigh, C. (2002) Using Chaos Theory: the implication for nursing, J. of Adv. Nurs., 37: 5,462-469.

Halberg, F., Cornélissen, G., Schack, B., Wendt, H.W., Minne, H., Sothern, R.B., Watanabe, Y., Katinas, G., Otsuka, K., Bakken, E.E. (2003) Blood Pressure Self-Surveillance for Health also Reflects 1.3-year Richardson Solar Wind Variation: Spin-off from Chronomics, Biomedicine & Pharmacotherapy. 57, 58s–76s.

Harre, R. (1985) The Philosophies of Science, Oxford, O.U.P., 2.

Lipscomb, P.A. (2003) Does Complex Adaptive Systems Theory Explain Psychotherapeutic Change? [Personal Communication].

Marks-Maran, D. (1999) Reconstructing nursing: evidence, artistry and the curriculum. Nurse Education Today, 19: 3–10.

Markus, M. (1992) Are one-dimensional maps of any use in ecology? Ecological Modelling, 63, 243-259.

May, R. (1989) The chaotic rhythms of life, New Scientist, 1691, 37-41.

Newell, D. (2003) Concepts in the study of complexity and their possible relation to chiropractic health care: a scientific rationale for a holistic approach, Clinical Chiropractic, 6:15-33.

Otero-Siliceo, E., Arriada-Mendicoa, N. (2003) Is it healthy to be chaotic? Medical Hypotheses. 60:2, 233–236.

Peitgen, H.O., Richter, P.H. (1986) The Beauty of Fractals, Berlin, Springer-Verlag.

Pesut, D.J. (2000) Care Emerges from Chaos, Nurs Outlook. 48:153.

Pickover, C.A. (1990) Computers, pattern, chaos and beauty, Stroud, Alan Sutton, 41.

Poston, T., Stewart, I. (1978) Catastrophe Theory and its applications, London, Pitman, 384.

Redington, D.J., Reidbord, S.P. (1992) Chaotic Dynamics in Autonomic Nervous System Activity of a Patient During a Psychotherapy Session. Biol. Psychiatry, 31:993-1007.

Rolf, G. (1999) The pleasure of the bottomless; postmodernism, chaos and paradigm shifts: A response to Di Marks-Maran Nurse Education Today (1999) 19(1): 3–11 Reconstructing nursing: evidence, artistry and the curriculum, Nurse Education Today, 19, 668–672.

Rosenhead, J. (1976) New Scientist, Prison Catastrophe, 15 July, 140.

Sarbadhikari, S.N., Chakrabarty, K. (2001) Chaos in the brain: a short review alluding to epilepsy, depression, exercise and lateralization, Medical Engineering & Physics. 23, 445–455.

Schroën, J.H.T. (2002) Non-linear dynamics and Chinese medicine: an essay on research models, TCM, and recent changes in modern scientific philosophy, Clinical Acupuncture and Oriental Medicine, 3, 92–98.

Thom, R. (1975) Structural Stability and Morphogenesis: An Outline of a General Theory of Models, Reading, Benjamin.

Thrift, N. (1999) Interdisciplinary application of nonlinear time series methods, Physics Reports, 308, 1–64.

Varela, F., Maturana, H., Uribe, R. (1974) Autopoiesis: the organisation of living systems, its characterization and a model. Biosystems 5:187-196.

Vicenzi, A.E. (1994) Chaos theory and some nursing considerations. Nurs. Science Quarterly, 7:1,36-42.

Vivaldi, F. (1989) New Scientist, 28.10.89, 1688, 49.

von Baeyer, H.C. (2003) Information: The New language of Science, London, Weidenfeld & Nicolson.

Walsh, M. (2000) Chaos, complexity and nursing. Nursing Standard, 14:32, 39-42.

Weidlich, W. (1991) Physics and Social Science: The Approach of Synergetics, Physics Reports, Review Section of Physics Letters, 204, 1,1—163.

Woodcock, A., Davis, M. (1980) Catastrophe Theory, Middlesex, Pelican Books.

Brooks, Myrna LaFleur. (1998). Exploring medical language: A student-directed approach (4th ed.) St. Louis: Mosby.

top : home