Welcome....

Okay, so, welcome to the blog that I set up for the Spring 2018 course AS.110.421 Dynamical Systems.  Really, this forum exists simply to allow me to throw out to you updates, random (but relevant) thoughts, take questions, and to extend lecture discussions.  Feel free to ask questions here, make comments, and interact as you see fit. 

Comments

Post a Comment

Popular posts from this blog

Lecture 4: Metrics, Lipschitz Continuity, and the Contraction Principle

The relevant text sections for this lecture are Section 2.1.3, 2.1.4, and 2.4.1.  Please read these prior to meeting on Thursday. Last lecture, we saw that understanding discrete dynamical systems will require an understanding of orbits and how they behave.  As discrete orbits are sequences in a space, convergence properties will be important.  One aspect of measuring convergence is via a metric, and metric spaces will play a vital role in this entire course.  In this lecture, we define a metric and discuss some of its properties.  We also introduce a notion of continuity, Lipschitz continuity, that will facilitate our study.  With that, we can move into our first major topic and theorem of the course:  The Contraction Principle.  While a straightforward concept, there are aspects of it that are a bit tricky, and we will take care to expose some of these thoughtful ideas as we discuss what is a contraction and what information does it convey....
Read more

Welcome to S20.AS.110.421 Dynamical Systems

Hey all,  So a couple of years back, I set up this blog to maybe create a forum for me and you guys to interact and discuss the current material from the course.  It was basically a means to attempt to go beyond the standard lecture/office hour/homework model.  Perhaps as you learn and explore, you will find things out on the internet that are relevant to the course.  Sharing them here would expose them to the entire class.  Also, questions about material is always best discussed as a group.  In any case, I will create a thread on each lecture, and also point to thing I find that are outside the text but interesting to think about or see.  You, in turn, are encouraged to participate, by posting comments, finding other examples of some of the topics of this course and sharing, or generally just getting involved.      I sincerely hope that you are already finding the material in, or the perspective of, this course interesting.  I ...
Read more

Lecture 15 - More Circle Maps: Toral flows, translations and circle homeomorphisms

Today, we continue with our characterization and classification of toral flows, but move into higher dimensions.  The results are pretty much the same, and again, rely on rigid rotations of each circle in the torus to characterize the flow.  But there is a twist (sorry for the pun) when we look at the time-1 map, an example of what we call a toral translation, or multi-dimensional rotation.  We will detail this today. Next, we will go back to simple circle maps, but move into a more general area of circle homeomorphisms rather than just rigid rotations.  It turns out that even here, there is a sort of rigidity (in that not much can happen) that is somewhat surprising.  For example, is a circle map has a n-periodic point, then it can only have possibly other points of period n, and no other period is possible other than n.  We will talk about why today.  The link is here .  See you today.
Read more