Hello, and welcome to Unit 10.
It's hard to believe but,
this is the last unit of the course.
So, there's not really any new material
in this course,
but I want to do a number of different things
to synthesize, summarize, and wrap up.
In, uh, this sub-unit, I'll do a topical summary of the course,
just a fairly quick reminder of the main results
uh, that we covered in the last nine weeks.
in the next two sub-units, I'll have interviews with, um,
two other researchers and scholars
the first is Stephen Keller, who's professor
of philosophy at Hamlin University
and he and I talked a bunch about a number
of philosophical issues
conceptual issues in chaos and dynamics
how is, um, chaos theory, uh,
a different sort of science
to what extend is it a different sort of science
than other areas
and some general thoughts about
interdisciplinary knowledge and interdisciplinary science.
Steven Morris, the second interview,
he's professor of geophysics at the University of Toronto
who's done a lot of experimental work in
nonlinear dynamic, and especially pattern formation
and we talked at some length about
some of his different experiments
and what draws him to study
pattern-forming systems
Then, I'll do a different sort of summary
that will be a little more conceptual and thematic
I want to draw out not so much the
technical results
but what I think some of the
big themes and lessons are
from dynamical systems,
and then finally, a few quick remarks
to wrap up the course.
So, let's get started with a topical overview
and summary
of what we've done the last nine weeks.
So let's review the main topics
from this course.
This course is about dynamical systems.
and a dynamical system
is a system that
evolves in time according to
a well-defined unchanging rule.
and the study of dynamical systems
as we've done in this course
is concerned with general properties of
dynamical systems.
We might seek to classify and characterize
the types of behavior
that we're seeing in dynamical systems.
there are other strands of
dynamical systems
that might focus on
particular areas of application
but in this introductory course,
we're taking a more general view
and just saying,
"What can dynamical systems do?"
and we looked at
two types of dynamical systems
iterated functions, and
differential equations.
so we started with iterated functions
and that's where
we start with an x-value
and we apply a function to it
and then we iterate that process
we repeat it
again and again and again
and we get a sequence of numbers
and we call that
the orbit, or the itinerary
So, it might be denoted this way
and the main example we studied was
the logistic equation
where the population of rabbits
on some mythical island
in the next year
is this particular function
of rabbits on the next year
uh um, in the current year
and maybe these should have n's on them
for um, completeness's sake.
fix that.
So, okay so we studied iterated functions
and we also studied differential equations
these are a little bit trickier mathematically
Here uh um the variable we're trying to
keep track of is
x and it's a function of time
but we're given it in terms of
a derivative.
The uh motivating example we looked at
was Newton's Law of Cooling
so if you place a cool beer in a warm room
it will the ... how the temperature changes
will be given by this formula
so this the rate of change of temperature