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To: yard_man who wrote (14862)	6/30/2006 3:23:53 PM
From: E. Charters	Respond to of 78419

The area of boundaries of different types of flow, disturbed flow, and turbulent flow creating complex harmonics is a admittedly difficult one. If one air mass flows at in a turbulent, disturbed fashion passing a threshold value and falling below it, then the equation becomes difficult and very non predictable because of its complexity or tendency to initiate harmonic oscillations or non repeating spiral cycles. For this reason, a pure flow program of differential probably cannot be simply implemented. One may have to combine statistics in order to arrive a modulating or F(d),U G{e} factors to control dependent variable escape. Studies on fluid flow in seemingly chaotic environments such as blood flow and turbulence in arteries, explaining plaque formation etc... has been attacked with some success by some investigators. These people say that there is hope that granularity solutions have efficacy. At least they have the experience to say so. It might be worth following up. "Systems behaviour, then, may be divided into two zones, plus the boundary between them. There is the stable zone, where if it is disturbed the system returns to its initial state; and there is the zone of instability, where a small disturbance leads to movement away from the starting point, which in turn generates further divergence. Which type of behaviour is exhibited depends on the conditions which hold: the laws governing behaviour, the relative strengths of positive and negative feedback mechanisms. Under appropriate conditions, systems may operate at the boundary between these zones, sometimes called a phase transition, or the ‘edge of chaos’. It is here that they exhibit the sort of bounded instability which we have been describing – unpredictability of specific behaviour within a predictable general structure of behaviour. Before the emergence of complexity theory, the unpredictability of such systems was attributed to randomness – a notion that bundles up all unexplained variation and treats it as best captured by probabilities. What actually happens on any given occasion is understood as the result of random choice among possible outcomes, but in proportion to their probabilities. Thus probability becomes a catch-all for what cannot be explained in terms of cause leading to effect; paradoxically the implication is that variation about predicted values results from as yet unexplained causal factors, and that as we understand more about what is going on the residual random element will be progressively reduced." "That last statement is not quite right. For though, the different streams of values outputted by the mathematical calculations, like the different weather sequences, are highly irregular, they are not formless. The indeterminate meanderings of these systems, plotted over time, show that there is a pattern to the movements. Though they are infinitely variable, the variation stays within a pattern, a family of trajectories. Such a pattern of trajectories (and a whole range of different ones have been identified by trying out interesting ideas in the branch of mathematics called topology) is called a strange attractor. They are called ‘strange’ to distinguish them from stable attractors, states to which the system reliably returns if disturbed. A strange attractor has the property of being fractal or self-similar – that is, its pattern repeats itself at whatever scale it is examined. Indeed one can say that chaos and fractals are mathematical cousins, with chaos emphasising the dynamics of irregularity, and fractals picking out its geometry (Stewart 1989). All of the above are, of course, ‘only’ abstract mathematical results – demonstrating at best that certain kinds of unstable behaviour are theoretically possible. However mathematicians are likely to assert that "anything that shows up as naturally as this in the mathematics has to be all over the place" (Stewart 1989, p.125). And the literature on chaos can cite examples that appear to validate this claim. One example is the wobbly orbit of Hyperion, one of Saturn’s planets. Another is the propagation of turbulence in fluids. In the field of chemistry, Prigogine and colleagues won the Nobel prize for work showing that under appropriate conditions chemical systems pass through randomness to evolve into higher level self-organised dissipative structures – so called because they dissipate unless energy is fed in from outside to maintain them. It has been used as the basis for an approach offering an alternative (or at least a complement) to Darwinian natural selection as an explanation of the ordered complexity of living organisms. And so on" "The consequences, as Stacey (1993) comprehensively summarises, are to turn much management orthodoxy on its head: analysis loses its primacy contingency (cause and effect) loses its meaning long-term planning becomes impossible visions become illusions consensus and strong cultures become dangerous statistical relationships become dubious. The list is longer." EC<:-}