CHAOS Todd Hutner

Background ► Classical physics is a deterministic way of looking at things. ► If you know the laws governing a body, we can predict where it will be at any given time thereafter. ► Classical physics also disregards nonlinearity. If the system behaves nonlinearly, find a linear approximation for it. ► That was the prevailing way of thinking about physics in 1960.

The Butterfly Effect The Butterfly Effect ► Edward Lorenz was an American Meteorologist. ► In the Early 1960’s, computer technology made solving complex systems of Differential Equations easier. ► Lorenz, in an attempt to better predict the Weather, had his computer solve a system of three nonlinear differential equations. ► Thus, given the data of the current weather, Lorenz could calculate the weather conditions far into the future.

The butterfly Effect ► Lorenz then wanted to extend the date of an earlier run. ► So, he started in the middle of the previous run. ► The assumption was that the computer would duplicate the earlier run. ► This, however is not what happened.

The Butterfly Effect ► Instead, the forecast diverged. Slowly at first, but eventually, the two weather patterns were extremely different. ► It turns out, the original value was , yet the value Lorenz fed into the computer for this subsequent run was rounded to.506. ► This led to a new idea regarding complex systems: sensitive dependence on initial conditions. This is why we can not predict the weather more then a few days in advance.

The Butterfly Effect ► Lorenz would eventually map his findings in phase space, giving rise to the Lorenz Attractor. ► ourse/Lesson1/Demos.html ourse/Lesson1/Demos.html ourse/Lesson1/Demos.html

Fractal geometry ► Benoit Mandelbrot was a Scientist at IBM in the 1960’s. ► The 1 st problem was to find any meaning from the fluctuations of Cotton Prices. ► Classical Economics predicted the ratio of small fluctuations to high ones to be very high. The data did not agree. ► Mandelbrot realized that while each price change was random, the curve for daily price change was identical to that for monthly, and yearly. ► This relationship held fast for over 60 years, including two world wars. ► This effect is called scaling. Patterns are the same on a large scale and a small scale, and all scales inbetween.

Fractal Geometry ► The second problem Mandelbrot encountered was so called “noise” in telephone lines used to transmit information between computers. ► The noise was random, but no mater how strong the signal, the engineers could never drown out the noise. ► Every so often, this noise would drown out a piece of information, creating an error.

Fractal Geometry ► According to the Engineers, there would be extended periods with out errors, followed by periods with. ► Mandelbrot realized you could break the periods with errors into smaller groups, some with errors, and some without. ► Mandelbrot then discovered that within any burst of errors, there would be smaller, error free periods. ► This relationship followed a scaling pattern, just as the cotton prices. The proportion of error free periods to error ridden periods was constant over any time scale.

Fractal Geometry ► This reminded Mandelbrot of the Cantor Set.

Fractal Geometry ► Mandelbrot then asked, what other shapes are similar. ► He came up with fractal geometry, a way to place an infinite line in a finite area. ► Fractal geometry gets its basis from the scaling principles he discovered in the two problems previously discussed.

Fractal Geometry

Bifurcation ► Robert May was interested in, among other things, the question of what happens to a populations as the rate of growth passes a critical point? ► The equation he was used to study this question was: x n+1 =rx n (1-x n ). In this case, r is rate of growth.

Bifurcation ► When r is low, the population tends toward extinction. ► As r increase, the population tends towards a single, steady state. ► As r continues to rise, the state splits, so that there are two steady states that the population oscillates between. ► As r continues to rise, the states split again, and again, into 4, 8, 16, etc. ► At some critical value of r, the population becomes chaotic, never tending toward a single steady state. ► The state of the population just before turning chaotic is called “self organized criticality.”

Bifurcation

Comets ► Most comets have an elliptical orbit, placing them just beyond the orbit of Jupiter. ► These orbits remain, for the most part, unchanged over time. ► Yet, a slight, new gravitational tug from Jupiter can send comets careening of into space, or plunge them into the inner solar system, flying by the orbit of Earth, and in toward the sun. ► This is another example of sensitive dependence to initial conditions.

Punctuated Equilibrium ► Darwin’s theory of Evolution describes natural selection as a gradual process, each individual difference building up over long periods of time. ► Yet, the fossil records shows long periods of stagnation followed by short periods of rapid change. ► Natural Selection alone can not account for this. ► We have seen how small changes initially can produce huge changes later on. ► So, a single organism adapting could cause larger scale adaptive changes over a short period of time.

The Extinction of the Dinosaurs ► Most scientist agree that a meteor striking the Earth was probably responsible for the extinction of the Dinosaurs. ► The commonly held belief is that it had to be a very large meteor to cause such a large extinction event. ► Yet, Chaotitians tend to disagree. They believe that the dinosaurs were in a self organized criticality. ► Once again, a very small change can drive this system into the chaotic realm of the bifurcation diagram.

Earth Processes ► The Earth, over time, has seen large scale fluctuations in its average global temperature, and its magnetic field. ► Most scientist are at a loss for an explanation why there are ice ages, as well as shifting magnetic fields. ► Chaotitians, on the other hand, believe that the Earth lies in the bifurcation diagram. Thus, the Earth can oscillate between the two (or more) values with out any reason.

Ventricle Fibrillations ► Fibrillation is where the heart no longer beats in a correct manner. This electrical pulse stimulates some parts of the heart to beat, while others do not. ► This can, if not corrected quickly, lead to death. ► Yet, after further investigation, it appears that no part of the heart is malfunctioning. Each cell works correctly on its own. ► But, the heart, as a whole, does not work.

Ventricle Fibrillations ► Chaotitians believe that there are two steady states to the heart. ► One is the correct beating pattern, and the other is fibrillation. ► Thus, it is believed that chaos will help us determine new, and improved ways to help patients going through ventricle fibrillations.

Conclusions ► Since the emergence of chaos, physics does not view nonlinearity as a problem. ► Also, physics does not believe it can predict the outcome of even the simplest of systems, to an infallible degree. ► Thus, nonlinear dynamic systems is one of the fastest growing fields in physics today.

Further Reading ► Chaos: Making a New Science by James Gleick ► The Essence of Chaos (The Jessie and John Danz Lecture Series) by Edward Lorenz ► Fractals and Chaos: The Mandelbrot Set and Beyond by Benoit Mandelbrot ► Stability and Complexity in Model Ecosystems by Robert May

Sources ► Gleick, James. Chaos: Making a New Science. Penguin Books, New York, ► Gribbin, John. Get a Grip on Physics. Ivy Press, East Sussex, 1999.