Nonlinear Systems

Outdated Paradigm

All good things come to an end and supremacy of the linear paradigm, characterised by utter certainty and predictability, was no exception. Einstein (1879-1955), Bohr (1885-1962), Schrödinger (1887-1961), Heisenberg (1901-1976) and Dirac (1902-1984) played a decisive role in pushing conventional wisdom within the natural sciences beyond the Newtonian limits that enveloped it centuries before. Later scholars working in different disciplines within the natural and life sciences demonstrated that uncertainty is an integral part of some phenomena; now described as being nonlinear.

In a reversal of the earlier trends at the beginning of the modern scientific era, discoveries in these sciences helped to initiate a process of change that is gradually introducing a measure of realism to expectations in the context of prediction and control of socio-economic events.

Nonlinear Phenomena

Nonlinear systems and processes do not present the familiar bell-shaped distribution associated with linear systems, where change is gradual and orderly and where measurements crowd together near an average value. On the contrary, Mandelbrot, and Gleick amongst others, discovered that in nonlinear systems change is more random and less predictable, and it involves discontinuities; rapid changes as opposed to smooth ones, and persistence; low for instance does not necessarily follow high.

Complex Systems

Over time, a group of nonlinear entities attracted particular interest. These systems are variously described as being complex, because they have numerous internal elements, dynamic, because their global behaviour is governed by local interactions between the elements, and dissipative, because they have to exchange energy with other systems to maintain stable self-organised global patterns. In addition, when the stable patterns are capable of evolution the systems are also depicted as being adaptive.

For the purpose of this website discussion of social, political and economic phenomena will focus exclusively on their behaviour as Complex Adaptive Systems.

Self-organised Complexity

Linear systems are found at or near equilibrium. A ball bearing on the rim of a bowl is a classic example; it quickly settles at the bottom and that is that. Nonlinearity, by contrast, is exhibited by systems that are far from equilibrium

Complex Adaptive Systems are able to assume stable global patterns although they exist in conditions that are far from equilibrium. This is a key feature that merits a few words here. The second law of thermodynamics states that when an organised system is left alone it drifts steadily into increasing levels of disorder. A deserted building, for instance, eventually turns into a pile of rubble. After a while even the rubble disappears without a trace. Ultimately, a system cut off from the outside world will fall into a state of randomised equilibrium in which little or nothing of interest ever happens. Basically, disorder is a more probable sate than order.

Put another way, for a system to remain in an organised stable state, it has to exchange (dissipate) energy, or matter, with other systems all the while. That is the only option open to it to avoid succumbing to the destructive power of the second law of thermodynamics. Without the nourishing rays of energy from the Sun, for instance, Earth would drift into complete equilibrium, and therefore nothingness. Continuous supply of energy from the Sun keeps the planet in a highly active state far from equilibrium. The energy is absorbed, dissipated and used to drive numerous local interactions that in total produce the stable pattern that we perceive as life on Earth. It should be noted that local chaotic agitation is necessary to produce an overall stable pattern. In other words, organised complexity emerges from a mix of chaos and order.

In brief, therefore, two key features are of major significance in understanding complexity:

  • A complex regime is a mix of global order and local chaos.
  • As long as local interactions proceed in a suitable manner the system will avoid chaos and will remain in a self-organised state of global order.

Complex Adaptive Systems

It is now possible to describe the traits that set Complex Adaptive Systems apart from other systems:

  • They have large numbers of internal elements that are lightly but not sparsely connected. The elements interact locally according to simple rules to provide the energy needed to maintain stable global patterns, as opposed to rigid order or chaos.
  • They have active internal elements that furnish sufficient local variety to enable the system to survive as it adapts to unforeseen circumstances. There are vast numbers of microstates inside the systems arising from numerous local interactions. There is, therefore, a high probability that at any time some of the microstates at least will find the prevailing conditions conducive to survival.
  • Variations in prevailing conditions result in many minor adaptations to the overall pattern of the system and a few large mutations, but it is not possible to predict the outcome in advance.
  • Predictability in Complex Adaptive Systems is limited to global patterns rather than the chaotic local details. Fundamentally, specific causes could not be linked to particular effects.

Evolution

There are also significant features that mark the way Complex Adaptive Systems evolve over time:

  • As Gell-Mann reported, at any point ‘complexity can either increase or decrease’, but ‘the greatest complexity represented has a tendency to grow larger with time.’
  • While the endless process of evolution unfolds there is a high probability that the average complexity of all systems will also increase.
  • Evolution normally involves relatively small adaptations that accumulate over long periods of time. There is no beginning or end and, hence, there is no idealised end-state to which the system progresses. Evolution in this case is a slow uphill marathon rather than a sprint to the nearest summit. Long-term survival is the main yardstick of success.
  • Evolution of Complex Adaptive Systems follows a punctuated equilibrium path. The global pattern of a Complex Adaptive System remains the same for relatively long periods of time and then undergoes fast radical change.

The interval of apparent inactivity mentioned above is deceptive; at the local level change takes place continuously and the system scrolls through many microstates within one global pattern, or attractor in complexity parlance.

Occasionally, however, relatively minor events succeed in shunting the system into another markedly different global pattern, or a new attractor. Only then does the observer recognise that a definite change had taken place, but in a stable system life quickly settles down to the new pattern. Variety, and hence flexibility, allows the system to bend with the wind, the only other alternative being extinction, the opposite of evolution.

Words of Caution

Complex and complexity as used in this website do not mean difficult or complicated. The terms apply to systems that are entirely different from the familiar linear systems encountered in Newtonian physics, as described above. It is necessary to underline this point in anticipation of a natural question: what is the use of pointing out the obvious when we all know that life is complex? This website, it is hoped, will demonstrate why it is essential to discuss the ‘obvious’.

Another word of caution is necessary. The emerging field of Complex Systems and Complexity has attracted many adherents from a variety of backgrounds and motivations. Some have made exaggerated claims as to what Complexity could offer. So for the record, Complexity will not toast the bread or make the beds of a morning. It certainly will not of its own change the world order and the global balance of power. It is simply a means to provide a better understanding of certain phenomena that have a radically different mode of behaviour from the more familiar mechanistic systems.

Further Reading

Coveney, P. and R. Highfield (1996) Frontiers of Complexity, London: Faber and Faber.
Elliot, E. and L. D. Kiel (eds.) (1997) Chaos Theory in the Social Sciences, Ann Arbor: University of Michigan Press.
Gell-Mann, M. (1994) The Quark and the Jaguar: Adventures in the Simple and Complex, Boston: Little, Brown and Company.
Gleick, J. (1988) Chaos: Making a New Science, London: Heinemann.
Gerogescu-Roegen (1971 but reprinted in 1999) The Entropy Law and the Economic Process, Cambridge (Mass.): Harvard University Press.
Kauffman, S. (1996) At Home in the Universe: the Search for Laws of Complexity, Oxford: Oxford University Press.
Kauffman, S. (1993) The Origins of Order, Oxford: Oxford University press.
Kravtsov, Y. A. and J. B. Kadtke (eds.) (1996) Predictability of Complex Dynamical Systems, Berlin: Springer-Verlag.
Lewin, R. (1997) Complexity: Life at the Edge of Chaos, London: Phoenix.
Nicolis, G. and Prigogine I. (1989) Exploring Complexity: An Introduction, New York: Freeman.
Rihani, S. (Spring 2002) Complex Systems Theory and Development Practice: Understanding Non-linear Realities, London: Zed Books.
Waldrop, M. M. (1994) Complexity, London: Penguin Books.