by Larry Bradley

2010

from ChaosAndFractals-TheSpaceTelescopeScienceInstitute Website

Weather prediction is an extremely difficult problem. Meteorologists can predict the weather for short periods of time, a couple days at most, but beyond that predictions are generally poor.

Edward Lorenz was a mathematician and meteorologist at the Massachusetts Institute of Technology who loved the study of weather. With the advent of computers, Lorenz saw the chance to combine mathematics and meteorology. He set out to construct a mathematical model of the weather, namely a set of differential equations that represented changes in temperature, pressure, wind velocity, etc.

In the end, Lorenz stripped the weather down to a crude model containing a set of 12 differential equations.

On a particular day in the winter of 1961, Lorenz wanted to re-examine a sequence of data coming from his model. Instead of restarting the entire run, he decided to save time and restart the run from somewhere in the middle. Using data printouts, he entered the conditions at some point near the middle of the previous run, and re-started the model calculation.

What he found was very unusual and unexpected. The data from the second run should have exactly matched the data from the first run. While they matched at first, the runs eventually began to diverge dramatically - the second run losing all resemblance to the first within a few "model" months.

A sample of the data from his two runs in shown below:

Lorenz's Sample Data
 

At first Lorenz thought that a vacuum tube had gone bad in his computer, a Royal McBee - an extremely slow and crude machine by today's standards.

After discovering that there was no malfunction, Lorenz finally found the source of the problem. To save space, his printouts only showed three digits while the data in the computer's memory contained six digits. Lorenz had entered the rounded-off data from the printouts assuming that the difference was inconsequential.

For example, even today temperature is not routinely measured within one part in a thousand.

This led Lorenz to realize that long-term weather forecasting was doomed. His simple model exhibits the phenomenon known as "sensitive dependence on initial conditions."

This is sometimes referred to as the butterfly effect, e.g. a butterfly flapping its wings in South America can affect the weather in Central Park. The question then arises,

Why does a set of completely deterministic equations exhibit this behavior?

After all, scientists are often taught that small initial perturbations lead to small changes in behavior. This was clearly not the case in Lorenz's model of the weather.

The answer lies in the nature of the equations; they were nonlinear equations.

While they are difficult to solve, nonlinear systems are central to chaos theory and often exhibit fantastically complex and chaotic behavior.

What is The Butterfly Effect?

from WiseGeek Website

The butterfly effect is a term used in Chaos Theory to describe how tiny variations can affect giant systems, and complex systems, like weather patterns.

The term butterfly effect was applied in Chaos Theory to suggest that the wing movements of a butterfly might have significant repercussions on wind strength and movements throughout the weather systems of the world, and theoretically, could cause tornadoes halfway around the world.

What the butterfly effect seems to posit, is that the prediction of the behavior of any large system is virtually impossible unless one could account for all tiny factors, which might have a minute effect on the system. Thus large systems like weather remain impossible to predict because there are too many unknown variables to count.

The term "butterfly effect" is attributed to Edward Norton Lorenz, a mathematician and meteorologist, who was one of the first proponents of Chaos Theory.

Though he had been working on the theory for some ten years, with the principal question as to whether a seagulls’ wing movements changes the weather, he changed to the more poetic butterfly in 1973.

A speech he delivered was titled, “Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas.” Actually, fellow scientist, Philip Merilees created the title. Lorenz had failed to provide a title for his speech.

The concept of small variations producing the butterfly effect actually predates science and finds its home in science fiction. Writers like Ray Bradbury were particularly interested in the types of problems that might occur if one traveled back in time, trailing anachronisms.

Could small actions taken in the past dramatically affect the future?

Fictional treatments of the butterfly effect as applies to time travel are numerous. Many cite the 2005 Butterfly Effect film as a good example of the possible negative changes that small behaviors in the past could have on the future, if one could time travel.

Actually, a better and more critically accepted treatment of this concept is the 2000 film Frequency. In the film a father and son communicate over time through radio waves and attempt to change the past for the good.

In human behavior, one can certainly see how small changes could render behavior, or another complex system, extremely unpredictable. Small actions or experiences stored in the unconscious mind, could certainly affect a person’s behavior in unexpected ways.

One looks at teen suicide for example, where no instance of previous depression has occurred. Loved ones are often left wondering what the many small factors were that precipitated a suicide. Further, people often agonize about the small details they did not see as possible factors for an unexpected suicide.

However, there are plenty of ways that such a behavior would be unanswerable according to the butterfly effect.

Minute actions and experiences dating from childhood stored in the unconscious mind are not accessible when a person has died, and they may be hard to access without hypnosis or therapy when a person is living.


 

The "Butterfly Effect"
July 10, 2003
from Templarser Website

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 Lorenz Attractor 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 trialed 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 behavior 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 modeled 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 modeling 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 ]

Butterfly Effects

Variations on a Meme
by Steve Nelson

June 03, 2008

from ClearNightSky Website

I found a file of search results I had compiled a long time ago as I was writing scripts to automatically scrape sites based on patterns. This experiment had to do with the Butterfly Effect, testing some regular expressions around "a butterfly flaps its wings in..."

The Butterfly Effect meme itself is invariant, but the variation in locations and effects is entertaining. My file of results notes that I did these searches using AltaVista - so that dates it a bit!

Here are a few of the results: