Rhythms in Chaos

TWO SELECTIONS FROM THIS CHAPTER.

Patterns in Chaos

The analysis of more complex systems, particularly following the development of high speed computers, has opened our eyes to a more diverse reality, which has a much richer fabric and more exciting dynamic. There are patterns in the chaos and the activities of life tend to be at the margins of chaos, where a meta-stability can be maintained through an open exchange of energy and a continual feedback of information. This understanding of more complex patterns has challenged the basic assumptions of the prevailing formulation of science. It suggests a world view different to that of classical European culture.

The analysis of system dynamics has shown that there are levels of stability: from the simple fixed stable point, to oscillation around an equilibrium condition, to the complex oscillations of the so called ‘strange attractors‘ of the theories of chaos. Regardless of the nature of the system, there is a well defined characteristic to these transitions from simple to complex stabilities. In the shift from periodic oscillations to the chaotic there is a pattern of doubling, of splitting of two cycle to four cycle to eight cycle etc, and this transition has the same form regardless of the specific nature of the dynamic system. Then, when these ‘strange attractors’, that define the range of variation of the system parameters, are plotted in a variable or state space (where the changing value of the variables is plotted) they display an interesting structure. The shape of the attractor plot is repeated at all scales. As any section of the plot is blown up, it displays an ever increasing detail, and at each level of detail there is a similar overall shape or structure.

This repeating pattern at all scales is a feature of what are called fractals, plots that are generated by a constant feedback, where the output from one iteration becomes the input for the next. Extraordinarily complex and beautiful structures can be generated in this way from the simplest of generator expressions.

This fractal repetition is evident throughout nature, in both physical and life processes, and through the high speed calculation capacity of computers very life like images can be generated from quite simple feedback equations. One example of this natural fractal repetition is in the shape of coastlines. The coast is a margin where earth, air and water interact, and erosion and deposition is generated by wind and water flows. This constant interplay gives rise to a spiral hook shape along the coast, and this shape is repeated at all levels. The same crenulate curve is present from the very large scale of a continental map, to the shape of beaches between headlines, to minor sandbars that move along the coast, to the detail of wave wash on the beach.

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The Complex Folding in of Feedback

In the theories of chaos, it is a folding in of the variable space that gives rise to the repeating patterns of the ‘strange attractors’, and hence the overall convergence or complex oscillation within the divergent ‘chaos’ of the processes. The essential principle of these dynamic systems is feedback, and it is the ability to monitor and adapt that allows systems to become self-organising entities. It requires openness and interactive relationships with other systems. Not the closed systems of controlled experiments.

The mathematical procedure that reflects this feedback is one of dependent relationships and continual iteration, where the result of a calculation is fed back as the input for the next calculation. A very simple demonstration of the shift in equilibrium states, from stable point to oscillation to complex attractors, is provided by an iteration of the parabolic relationship. A responsive feedback system must have a contained ‘solution space’, and in geometric terms that means the graph must fold back on itself. The simplest of curved graphs is a parabola, of the form y = x2. For a given range of the parabola of 0 to 1 and a function variable k (which defines the parabola height) the mathematical expression takes the form of: y = 4kx (1-x).

If the line x = y represents the conversion of the output to the input of the iteration, the series of values generated by the iteration of the parabolic relationship can be represented graphically by a line that goes back and forth between a 45° line (of x = y) and the parabola – as shown opposite.

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