Barbara Perez
Chaos and the Iteration of the Saw-Tooth Function
Lecture 2/14/07
Assignment #1 - Find Ed Lorenz on the internet. Write a few pages about him and
his work on chaos
Edward Norton Lorenz
Edward Norton Lorenz was born in 1917. He is a mathematician and
meteorologist. He is known as a pioneer of chaos theory. He
invented the strange attractor notion and coined the term butterfly
effect.
He was born in West Hartford, Connecticut, and attended Dartmouth
College and Harvard University.
During World War II, he served as a weather forecaster for the
United States Army Air Corps. Upon his return from the war, he
studied meteorology at MIT and later became a professor there.
Lorenz’s interest in chaos theory came about accidentally through his work on
weather prediction in 1961. As Lorenz studied weather patterns he began to realize
that the weather did not always change as predicted. During two runs of an
identical simulation he realized that the prediction results were much different. He
investigated and found out that his data was rounded to a three-digit number and
the computer he was using worked with six-digit numbers. Although the
difference was small, the difference in the outcome was large. This sensitive
dependence on initial conditions came to be known as the butterfly effect, because
of the title of a paper by Edward Lorenz in 1972 entitled Predictability: Does the
Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The flapping
wing represents a small change in the initial condition of the system, which causes
a chain of events leading to large-scale phenomena. Had the butterfly not flapped
its wings, the trajectory of the system might have been vastly different. With the
realization that seemingly simple systems may be unexpectedly complex
mathematically, Edward Lorenz, changed the way scientists look at the world.
In mathematics, chaos theory describes the behavior of certain systems that exhibit
certain characteristics. One of these characteristics is sensitivity to initial
conditions. This characteristic can also be called, using Lorenz’s term, the
“butterfly effect.” As a result sensitivity, the behavior of systems that exhibit
chaos appears to be random. Some examples of chaotic systems include the
atmosphere, the solar system, plate tectonics, turbulent fluids, economics, and
population growth.
Lorenz built a mathematical model of the way air moves around in the
atmosphere. He discovered the Lorenz equations in 1963 as a very simplified
model of convection rolls in the upper atmosphere.
In his work, Lorenz described this system of equations, which
resulted in what is known as the Lorenz attractor. His published
work describing this system of equations is titled “Deterministic
Nonperiodic Flow” in the Journal of Atmospheric Sciences.
According to his analysis, the solutions to the equations can
only be predicted for a finite period of time. For weather
phenomena, the limit of predictability is about two days.
Lorenz found that the trajectories of this system, for certain
settings, never settle down to a fixed point, never approach a
stable limit cycle, yet never diverge to infinity. What Lorenz
discovered was at the time unheard of in the mathematical
community, and was largely ignored for many years. The
Lorenz attractor is a chaotic map, noted for its butterfly shape.
Now this beautiful attractor is the most well known strange
attractor that chaos has to offer.
A simple physical model of the Lorenz equations at work is a
leaky waterwheel. A waterwheel built from paper cups with equal
sized holes in the bottom of each cup is allowed to turn freely
under the force of a steady stream of water poured into the top
cup. For a slow flow of water, the water leaks out fast enough that
friction keeps the waterwheel from moving. For just a little more
flow the waterwheel will pick a direction and spin in that
direction forever. If the flow is increased further the waterwheel
does not settle into a stable cycle. Instead it spins in one direction
for a bit, then slows down and starts to spin the other. The
waterwheel will constantly change its direction of spin, and never
in a repeating predictable manner.
A plot of the trajectory
Lorenz system.
Edward Lorenz's work in chaos theory changed the scientific
fields of meteorology, biology and fluid mechanics. Kevin Cuomo and Alan
Oppenheim have built on the work of Pecora and Carrol's synchronized chaos.
They apply the Lorenz equations to data encryption.
Sources
http://www.apmaths.uwo.ca/~bfraser/nll/version1/lorenzintro.html
http://en.wikipedia.org/wiki/Lorenz_system
http://en.wikipedia.org/wiki/Chaos_theory
http://en.wikipedia.org/wiki/Edward_Lorenz
http://www.psu.edu/ur/archives/intercom_1996/April25/CURRENT/lorenz.html
Assignment #2- In the Java Applet use the button 1 to activate the synchronous iteration
of ax(1-x) and ax-ax^2. Field 3 permits you to choose values for a and the Checkbox 2
allows you to have the difference displayed. Play with some values of a and observe
whether or not the difference eventually becomes non-zero. Document your findings and
when you have an a-value for which the difference remains at zero, check graphical
iterations for that a-value and compare with graphical iterations for a-values with nonzero difference.
The lowest number that I found to eventually have a non-zero difference was 3.75.
Anything greater than that also eventually had a non-zero difference. Now once you
compare the graphical iterations for the a-values that produce a non-zero difference vs. avalues with a constant zero difference, you can see an obvious distinction. Those avalues with a non-zero difference, such as 3.75 and 4, have an erratic graph that goes up
and down between 0 and 1. The a-values with a zero difference, such as 1 or 2, produce a
graph which is reaching some limit.
a=3.75
a=2
a=4
a=1
Assignment #3 - In the Java Applet use the button 1 to activate the iteration of ax(1-x).
Field 3 permits to choose a. Now choose two initial values which are 1/10, or 1/100, or
1/1000, or 1/10000 apart. Checkbox 2 allows you to have the difference of the two
synchronous iterations displayed.
Now play with values of a and observe whether or not the difference eventually becomes
large (>0.5).
Document your findings and when you have an a value for which the difference remains
small, check graphical iterations for that a value and compare with graphical iterations
for a values where the difference becomes large.
With the parameter set at 3.99, and initial values of .4 and .5, the difference becomes >0.5
very quickly. With the parameter set at 3.99, and initial values of .5 and .51, the
difference also becomes >0.5 very quickly. With the parameter set at 3.99, and initial
values of .5 and .501, the difference becomes >0.5, but it takes longer than when the
initial difference was larger. With the parameter set at 3.99, and initial values of .5 and
.5001, the difference becomes >0.5, but it takes longer than when the initial difference
was larger. With the parameter set at 3.99, and initial values of .5 and .50001, the
difference becomes >0.5, but it takes longer than when the initial difference was larger. If
you change the parameter to 2.0, and keep the same initial values, the difference stays
small eventually becoming zero regardless of the difference of the initial values.
In trying different parameters, 3.75 is the lowest value of a for which the difference
becomes >0.5. Anything less than 3.75, the difference is small and eventually zero.
The following are comparisons of graphical iterations for a-values where the difference
stays small to a-values where the difference becomes large. The graphs on the left are the
a-values where the difference stays small. As you can see these graphs are predictable
and uniform. The graphs on the right are a-values where the difference is large. These
graphs are erratic and unpredictable.