The Butterfly Effect

The "Butterfly Effect" is the propensity of a system to be sensitive to initial conditions.Such systems over time become unpredictable,this idea gave rise to the notion of a butterfly flapping it's wings in one area of the world,causing a tornado or some such weather event to occur in another remote area of the world.

 Comparing this effect to the domino effect,is slightly misleading.There is dependence on the initial sensitivity,but whereas a simple linear row of dominoes would cause one event to initiate another similar one,the butterfly effect amplifies the condition upon each iteration.

 The butterfly effect has been most commonly associated with the Weather system as this is where the discovery of "non-linear" phenomenon began when Edward Lorenz found anomalies in computer models of the weather.But Henri Poincaré had already made inroads into this area. Mapping the results in "phase space" produced a two-lobe map called the Lorenz Attractor. The word attractor meaning that events tended to be attracted towards the two lobes,and events outside of the lobes are such things like snow in the desert.

 The attractor acts like an egg whisk,teasing apart parameters that may initially be close together,this is why the weather is so hard to predict. Super computers run several models of the weather in parallel to discover whether they stay close together or diverge away from each other.Models that stay similar in nature give an indication that the weather is relatively predictable,and are used to indicate the confidence level that Meteorologists have in a prediction.

 It is not just the weather though that is subject to such phenomena.Any "Newtonian Classical" system where one system is in competition with another,such as the "Chaotic Pendulum" which plays magnetism off against gravity will exhibit "sensitivity to initial conditions".

 Animal populations may also be subject to the same phenomena.Work done by Robert May,suggests that predator-prey systems have complex dynamics making them prone to "boom" and "bust",due to the difference equations that model them.Such a system even with two variables such as Rabbits and Foxes can create a system that is much more complex than would be thought to be the case.Lack of  Foxes means that the Rabbit population can increase,but increasing numbers of Rabbits means Foxes have more food and are likely to survive and reproduce,which in turn decreases the number of Rabbits.It is possible for such systems to find a steady state or equilibrium,and even though species can become extinct,there is a tendency for populations to be robust,but they can vary dramatically under certain circumstances. Real populations of course,have more than two variables making them ever more complex.But as can be seen from the diagram, such systems are not as simple as might be thought.

The chemical world is also not free from such intrusions of non-linearity.In certain cases chemical feedback produces effects as that in the Belousov-Zhabotinsky reaction, creating concentric rings, which are produced by a chemical change, whose decision to change from one state to another cannot be predicted.The B-Z chemical system is currently being trialled as a means to achieve artificially intelligent states in robots.
Phase space portraits of liquid flow show that they too are subject to the same kind of non-linearity that is inherent in other physical systems.It may be apparent when turning on a tap that sporadic drips become "laminar" as the flow increases.What might not be apparent is the nature of the change from semi-random to continuous.It may seem rather at odds with intuition that such natural systems have inherent behaviour that is not random,or indeed that is not capable of being predicted.It may also seem that "not random" means "predictable".

Natural systems can present a tangled mix of determinism and randomness,or "order" and "chaos".In such cases as water moving from drips to continuous flow, pictures called "Bifurcation diagrams" demonstrate the nature of movement from order into chaos.This bifurcation is based on Robert May's work,but one of the intriguing things about bifurcations is that the same pattern occurs no matter what system is iterated.In fact Mitchell Feigenbaum discovered that there was a "constant of doubling" hidden in amongst all these systems.

 Electronic apparatus is also not free from such effects,and it is perhaps ironic,that we think of electronic apparatus as as being the epitome of predictable determinism and ruthless clockwork efficiency.Indeed the powerful computers used to predict weather,would seem ineffectual if they were not ruthless automatons.But such effects occur only in certain circumstances where there is "sensitivity to initial conditions".Amplifiers for instance,produce a howl when feedback occurs as they go into a stable state of oscillation.Logic gates as used in computers have to select a "0" or a "1",and this relies on choosing between two states whose boundary is indeterminate,and it is when a computer confuses a "0" for a "1" or vice versa that mistakes occur.

Phase space portrait of regular flow in a Taylor-Couette system	Phase space portrait of chaotic flow as the attractor becomes "strange"

Phase space portraits of a system of coupled pendulums

Many of the shapes that describe non-linear systems are fractal,a set of shapes that are self-similar on smaller and smaller scales with no limit to the size of the scale. Fractals were discovered by Benoit Mandelbrot at IBM.

A picture of a real fern and an affine transformation performed by "Winfract" (Inset1).Inset 2 is a real image of a conifer,and bears an uncanny resemblance to a diffusion limited aggregate.

 Fractals have been seen as describing naturally occurring phenomena such as the cragginess of mountains or the shapes of certain plant forms, such as ferns,which can be modelled by affine transformations.

Whether in fact Nature is fractal,or whether it just describes it better than the simple geometry of Euclid depends on the philosophical view taken of mathematics as a whole. Some people think mathematics is just a tool or a creation of man,and therefore Nature is only described or mapped by mathematics.

Others think that the description is real -at least in the sense that the similarity is not superficial,that in fact natural objects that look fractal,or which fractals look like,are similar in appearance because at some fundamental level the natural objects are obeying some form of rule system that bears a similarity to the sort of rules which govern fractals.

Whichever way you look at it,one thing no one can say is that mathematics is irrelevant to Nature. From butterflies to plants,from the weather to chemistry,mathematics is modelling or displaying attributes of Nature,and helping us to understand what we see.

A new conception was being made....that whatever fundamental units the world is put together from,they are more delicate,more fugitive,more startling than we catch in the Butterfly Net of our senses.
Jacob Bronowski "The Ascent of Man"

How to Float Like a Butterfly The intricacies of insect flight are astounding. But the animals' small size and swift movements make detailed studies of their aerodynamic acrobatics difficult. Now results of a study of free-flying butterflies published today in the journal Nature suggest that the insects rely on a variety of techniques, often employed in successive strokes, over the course of a flight. R. B. Srygley and Adrian L.R. Thomas of the University of Oxford trained red admiral butterflies to fly toward a fake flower at the end of a wind tunnel. Photographs snapped as the insects moved through wisps of smoke in the chamber provided the researchers with an opportunity to analyze their flight patterns. Specifically, the scientists matched patterns in the airflow around the butterflies' wings to standard mathematical patterns with known properties. They determined that the creatures use a number of "unconventional aerodynamic mechanisms to generate force." What is more astounding, writes Rafal Zbikowski of Cranfield University in an accompanying commentary, is that "the butterflies appear to switch effortlessly among these mechanisms from stroke to stroke." Indeed, he concludes, if engineers ever succeed in understanding just how insects exercise control over such a wide range of abilities, "there will be a revolution in aeronautics." --Sarah Graham [Scientific American December 12, 2002 ] Theology and Chaos | The Self Organizing Universe  | MP3 130K MP3 214K