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