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

CHAOS CHAOS Todd HutnerTodd Hutner

BackgroundBackground

► Classical physics is a deterministic way of Classical physics is a deterministic way of looking at things. looking at things.

► If you know the laws governing a body, we If you know the laws governing a body, we can predict where it will be at any given can predict where it will be at any given time thereafter. time thereafter.

► Classical physics also disregards Classical physics also disregards nonlinearity. If the system behaves nonlinearity. If the system behaves nonlinearly, find a linear approximation for nonlinearly, find a linear approximation for it. it.

► That was the prevailing way of thinking That was the prevailing way of thinking about physics in 1960. about physics in 1960.

The Butterfly EffectThe Butterfly Effect

► Edward Lorenz was an American Edward Lorenz was an American Meteorologist.Meteorologist.

► In the Early 1960’s, computer technology In the Early 1960’s, computer technology made solving complex systems of made solving complex systems of Differential Equations easier.Differential Equations easier.

► Lorenz, in an attempt to better predict the Lorenz, in an attempt to better predict the Weather, had his computer solve a system Weather, had his computer solve a system of three nonlinear differential equations. of three nonlinear differential equations.

► Thus, given the data of the current weather, Thus, given the data of the current weather, Lorenz could calculate the weather Lorenz could calculate the weather conditions far into the future. conditions far into the future.

The butterfly EffectThe butterfly Effect

►Lorenz then wanted to extend the date Lorenz then wanted to extend the date of an earlier run. of an earlier run.

►So, he started in the middle of the So, he started in the middle of the previous run. previous run.

►The assumption was that the The assumption was that the computer would duplicate the earlier computer would duplicate the earlier run. run.

►This, however is not what happened. This, however is not what happened.

The Butterfly EffectThe Butterfly Effect

► Instead, the forecast diverged. Slowly at first, Instead, the forecast diverged. Slowly at first, but eventually, the two weather patterns were but eventually, the two weather patterns were extremely different. extremely different.

► It turns out, the original value was .506127, It turns out, the original value was .506127, yet the value Lorenz fed into the computer for yet the value Lorenz fed into the computer for this subsequent run was rounded to .506.this subsequent run was rounded to .506.

► This led to a new idea regarding complex This led to a new idea regarding complex systems: sensitive dependence on initial systems: sensitive dependence on initial conditions. This is why we can not predict the conditions. This is why we can not predict the weather more then a few days in advance. weather more then a few days in advance.

The Butterfly Effect The Butterfly Effect

►Lorenz would eventually map his Lorenz would eventually map his findings in phase space, giving rise to findings in phase space, giving rise to the Lorenz Attractor. the Lorenz Attractor.

►http://www.cmp.caltech.edu/~mcc/Chahttp://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demos.htmlos_Course/Lesson1/Demos.html

Fractal geometryFractal geometry

► Benoit Mandelbrot was a Scientist at IBM in the 1960’s.Benoit Mandelbrot was a Scientist at IBM in the 1960’s.► The 1The 1stst problem was to find any meaning from the problem was to find any meaning from the

fluctuations of Cotton Prices. fluctuations of Cotton Prices. ► Classical Economics predicted the ratio of small Classical Economics predicted the ratio of small

fluctuations to high ones to be very high. The data did fluctuations to high ones to be very high. The data did not agree.not agree.

► Mandelbrot realized that while each price change was Mandelbrot realized that while each price change was random, the curve for daily price change was identical random, the curve for daily price change was identical to that for monthly, and yearly. to that for monthly, and yearly.

► This relationship held fast for over 60 years, including This relationship held fast for over 60 years, including two world wars. two world wars.

► This effect is called scaling. Patterns are the same on a This effect is called scaling. Patterns are the same on a large scale and a small scale, and all scales inbetween. large scale and a small scale, and all scales inbetween.

Fractal GeometryFractal Geometry

►The second problem Mandelbrot The second problem Mandelbrot encountered was so called “noise” in encountered was so called “noise” in telephone lines used to transmit telephone lines used to transmit information between computers.information between computers.

►The noise was random, but no mater The noise was random, but no mater how strong the signal, the engineers how strong the signal, the engineers could never drown out the noise.could never drown out the noise.

►Every so often, this noise would drown Every so often, this noise would drown out a piece of information, creating an out a piece of information, creating an error. error.


► According to the Engineers, there would be According to the Engineers, there would be extended periods with out errors, followed by extended periods with out errors, followed by periods with. periods with.

► Mandelbrot realized you could break the periods Mandelbrot realized you could break the periods with errors into smaller groups, some with errors, with errors into smaller groups, some with errors, and some without. and some without.

► Mandelbrot then discovered that within any burst of Mandelbrot then discovered that within any burst of errors, there would be smaller, error free periods. errors, there would be smaller, error free periods.

► This relationship followed a scaling pattern, just as This relationship followed a scaling pattern, just as the cotton prices. The proportion of error free the cotton prices. The proportion of error free periods to error ridden periods was constant over periods to error ridden periods was constant over any time scale. any time scale.


► This reminded Mandelbrot of the Cantor Set. This reminded Mandelbrot of the Cantor Set.


►Mandelbrot then asked, what other Mandelbrot then asked, what other shapes are similar.shapes are similar.

►He came up with fractal geometry, a He came up with fractal geometry, a way to place an infinite line in a finite way to place an infinite line in a finite area. area.

►Fractal geometry gets its basis from the Fractal geometry gets its basis from the scaling principles he discovered in the scaling principles he discovered in the two problems previously discussed. two problems previously discussed.


BifurcationBifurcation

►Robert May was interested in, among Robert May was interested in, among other things, the question of what other things, the question of what happens to a populations as the rate happens to a populations as the rate of growth passes a critical point? of growth passes a critical point?

►The equation he was used to study this The equation he was used to study this question was: xquestion was: xn+1n+1=rx=rxnn(1-x(1-xnn). In this ). In this case, r is rate of growth. case, r is rate of growth.


► When r is low, the population tends toward When r is low, the population tends toward extinction. extinction.

► As r increase, the population tends towards a As r increase, the population tends towards a single, steady state. single, steady state.

► As r continues to rise, the state splits, so that there As r continues to rise, the state splits, so that there are two steady states that the population oscillates are two steady states that the population oscillates between. between.

► As r continues to rise, the states split again, and As r continues to rise, the states split again, and again, into 4, 8, 16, etc. again, into 4, 8, 16, etc.

► At some critical value of r, the population becomes At some critical value of r, the population becomes chaotic, never tending toward a single steady state. chaotic, never tending toward a single steady state.

► The state of the population just before turning The state of the population just before turning chaotic is called “self organized criticality.” chaotic is called “self organized criticality.”


CometsComets

►Most comets have an elliptical orbit, placing Most comets have an elliptical orbit, placing them just beyond the orbit of Jupiter. them just beyond the orbit of Jupiter.

► These orbits remain, for the most part, These orbits remain, for the most part, unchanged over time. unchanged over time.

► Yet, a slight, new gravitational tug from Yet, a slight, new gravitational tug from Jupiter can send comets careening of into Jupiter can send comets careening of into space, or plunge them into the inner solar space, or plunge them into the inner solar system, flying by the orbit of Earth, and in system, flying by the orbit of Earth, and in toward the sun. toward the sun.

► This is another example of sensitive This is another example of sensitive dependence to initial conditions. dependence to initial conditions.

Punctuated EquilibriumPunctuated Equilibrium

► Darwin’s theory of Evolution describes natural Darwin’s theory of Evolution describes natural selection as a gradual process, each individual selection as a gradual process, each individual difference building up over long periods of time.difference building up over long periods of time.

► Yet, the fossil records shows long periods of Yet, the fossil records shows long periods of stagnation followed by short periods of rapid stagnation followed by short periods of rapid change. change.

► Natural Selection alone can not account for this.Natural Selection alone can not account for this.► We have seen how small changes initially can We have seen how small changes initially can

produce huge changes later on.produce huge changes later on.► So, a single organism adapting could cause So, a single organism adapting could cause

larger scale adaptive changes over a short larger scale adaptive changes over a short period of time.period of time.

The Extinction of the The Extinction of the DinosaursDinosaurs

►Most scientist agree that a meteor striking Most scientist agree that a meteor striking the Earth was probably responsible for the the Earth was probably responsible for the extinction of the Dinosaurs. extinction of the Dinosaurs.

► The commonly held belief is that it had to be The commonly held belief is that it had to be a very large meteor to cause such a large a very large meteor to cause such a large extinction event. extinction event.

► Yet, Chaotitians tend to disagree. They Yet, Chaotitians tend to disagree. They believe that the dinosaurs were in a self believe that the dinosaurs were in a self organized criticality. organized criticality.

►Once again, a very small change can drive Once again, a very small change can drive this system into the chaotic realm of the this system into the chaotic realm of the bifurcation diagram. bifurcation diagram.

Earth ProcessesEarth Processes

► The Earth, over time, has seen large scale The Earth, over time, has seen large scale fluctuations in its average global fluctuations in its average global temperature, and its magnetic field. temperature, and its magnetic field.

►Most scientist are at a loss for an Most scientist are at a loss for an explanation why there are ice ages, as well explanation why there are ice ages, as well as shifting magnetic fields. as shifting magnetic fields.

► Chaotitians, on the other hand, believe that Chaotitians, on the other hand, believe that the Earth lies in the bifurcation diagram. the Earth lies in the bifurcation diagram. Thus, the Earth can oscillate between the Thus, the Earth can oscillate between the two (or more) values with out any reason. two (or more) values with out any reason.

Ventricle Fibrillations Ventricle Fibrillations

► Fibrillation is where the heart no longer Fibrillation is where the heart no longer beats in a correct manner. This electrical beats in a correct manner. This electrical pulse stimulates some parts of the heart to pulse stimulates some parts of the heart to beat, while others do not.beat, while others do not.

► This can, if not corrected quickly, lead to This can, if not corrected quickly, lead to death.death.

► Yet, after further investigation, it appears Yet, after further investigation, it appears that no part of the heart is malfunctioning. that no part of the heart is malfunctioning. Each cell works correctly on its own. Each cell works correctly on its own.

► But, the heart, as a whole, does not work. But, the heart, as a whole, does not work.

Ventricle Fibrillations Ventricle Fibrillations

►Chaotitians believe that there are two Chaotitians believe that there are two steady states to the heart.steady states to the heart.

►One is the correct beating pattern, and One is the correct beating pattern, and the other is fibrillation. the other is fibrillation.

►Thus, it is believed that chaos will help Thus, it is believed that chaos will help us determine new, and improved ways us determine new, and improved ways to help patients going through to help patients going through ventricle fibrillations. ventricle fibrillations.

ConclusionsConclusions

►Since the emergence of chaos, physics Since the emergence of chaos, physics does not view nonlinearity as a problem.does not view nonlinearity as a problem.

►Also, physics does not believe it can Also, physics does not believe it can predict the outcome of even the simplest predict the outcome of even the simplest of systems, to an infallible degree. of systems, to an infallible degree.

►Thus, nonlinear dynamic systems is one Thus, nonlinear dynamic systems is one of the fastest growing fields in physics of the fastest growing fields in physics today. today.

Further Reading Further Reading

►Chaos: Making a New Science Chaos: Making a New Science by James by James GleickGleick

►The Essence of Chaos (The Jessie and The Essence of Chaos (The Jessie and John Danz Lecture Series)John Danz Lecture Series) by Edward by Edward LorenzLorenz

►Fractals and Chaos: The Mandelbrot Set Fractals and Chaos: The Mandelbrot Set and Beyondand Beyond by Benoit Mandelbrot by Benoit Mandelbrot

►Stability and Complexity in Model Stability and Complexity in Model Ecosystems Ecosystems by Robert Mayby Robert May

Sources Sources

►Gleick, James. Gleick, James. Chaos: Making a New Chaos: Making a New Science.Science. Penguin Books, New York, Penguin Books, New York, 1987. 1987.

►Gribbin, John. Gribbin, John. Get a Grip on Physics.Get a Grip on Physics. Ivy Press, East Sussex, 1999. Ivy Press, East Sussex, 1999.

Documents

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