I am Valerie Bodell. Welcome to, "What has
Chaos Done for Me Lately?" This course is about
applying chaos theory to the real world and it is a
five week, one credit course. As promised there are no books to buy,
you don't have to work any math or physics problems, unless you want to, of
course.
The syllabus is in d2l and you can read it there,
but there's a couple things I wanted to tell you about in particular. One is that
discussions are large part of your grade.
In order to get full credit on discussions, you must participate in the
conversation,
not just to reiterate what your classmates have written.
Consider composing your post before you read what they've written
and then putting it on d2l. Replying with,
"I agree," will not be enough. Consider
commenting on what your classmates have written. Maybe taking it a step further.
Maybe bringing up something new. Something like that would be fine.
The best way to contact me is by email.
I generally respond to emails within 24 hours,
but I check my email less often on the weekends,
so please do not wait to the last minute to read your assignment and realize that
you don't know what you're supposed to do.
My contact information is on the syllabus,
in d2l and in the overview of the course. It's bodell@mail.etsu.edu.
I also want to let you know that your final assignment is a formal paper.
Five to seven pages. 1000 to 1500 words
on any topic that interests you relating to the chaos theory.
In each week's assignments I've included some
optional readings. I'm hoping that some of those readings will correspond to
your interests and you can use them as a jumping-off point for your paper,
a place to get references, something like that. I've also put a few tips
on getting information from the Sherrod Library,
maybe a couple other resources in there. Just check out each week
and you'll find stuff that'll, hopefully help you. This class
is actually the second in a trilogy of related courses about chaos.
If you happened to take the first course, Order in Chaos,
then this next part will be in review for you. If you didn't take the first course,
no worries, these classes are independent of each other. This next part will be a
brief summary
of things that will really help you understand the readings
and writing your paper and just give you an idea about chaos.
So first let me start with six characteristics of
chaotic systems that will be important for this course.
Number one: They are dynamical, which means they change through time.
The quintessential example of a chaotic system
is the weather and we all know it changes through time.
The second characteristic: chaotic systems are produced by
iteration. That means the repetition of a process.
In our example of the weather, of course, we repeat days and nights following each other
as the earth rotates
and as the Earth revolves around the Sun we repeat seasons in years.
Third characteristic: chaotic systems are sensitive to initial conditions.
This means that if you had two processes that start from
infinitesimally different initial conditions,
they will reach wildly different ends.
In our weather example, this explains why
our predictions are so unworthy.
Because the weather is such a complex system. It has so many variables
and we cannot define those variables exactly enough
to get our prediction the same as the weather actually is.
So very quickly our forecast verges
from reality. The fourth characteristic
is chaotic systems are non linear.
This is similar to being sensitive to initial conditions.
It just means that there's no simple
arithmetic relationship between the variables and time.
So a small change does not have a proportionally small
outcome. The outcome can be wildly different.
The fifth characteristic
is that chaotic systems are deterministic. This means
that if you had two processes that you could
start from the exact same initial conditions, they would reach
the same ends. So the outcome depends on the initial conditions.
Finally, the sixth characteristic is that chaotic systems are self-similar
at different scales. Not self-same,
but self-similar at different scales, so a mountain is similar
to the large mountain range, the branching veins in a leaf are similar
to the branching bowels on whole tree and a single summer day
similar to the whole summer season.
Chaotic systems are so complex and the mathematics
so unwieldy, that scholars often use graphic
representations to help us understand the system.
They often use fractal to represent chaotic systems.
So chaotic systems can be represented by fractals,
but not all fractal represent chaotic system.
Fractals refer to fractional dimensions.
I think we all know what fractions are; something less than a whole number,
one-half, three-quarters. And dimensions, we probably are also,most of us, familiar
with.
A line has one dimension. A plane
has two dimensions. The area inside the shape
has three dimensions and space-time has four dimensions.
So let's consider the steps in the creation of the Hilbert curve.
As you can see this shape, which is called a curve just because it's not a
straight line,
is formed by the repetition a simple shapes
and you can imagine if you continue this process for ever
this line would get longer and longer and more and more complicated.
And it would eventually fill up all of the space in a plane.
So one line, which normally has one dimension,
would eventually fill the space in a plane, which has two dimensions.
So the Hilbert curve has two dimensions.
Now let's consider a shape that has a fractional dimension.
Let's look at the steps in the creation of the Koch
snowflake. This shape is, again, formed by the repetition of a simple shape
and as we repeat the shape, the one line gets fatter and fuzzier,
but it will never fill up all of the space in a plane. It remains in the
snowflake shape,
so it takes up more space but it's still not
two-dimensional. The Koch snowflake as a dimension of 1.26186.
Well this gives you some idea about chaos
and some ways to visualize it. In this first week,
I want you to think about how complex our world is.
This course is relating chaos theory to our world.
So think about the fact that there's no pristine place left on earth.
There's no place untouched by human air pollution
or oil spills or economic or political machinations.
It's not just the world, think about all the influences on your life.
How complex it would be to try to account for them all.
And what role has serendipity or chance played?
This is a lot to think about. Take your time to think about it. Think about what
interests you.
And I look forward to working with you over the next weeks. Thank you.