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