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. 

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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....
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Lecture 12 - Linear Maps of the plane and the Matrix Exponential

So for tomorrow, we will finish classifying the orbit structure of linear maps of the plane, basically concluding that orbits of planar linear maps follow similar patterns as that found in the trajectories of linear, homogeneous, autonomous ODEs in 2-dimensions. But there is a twist, in that it is the sign of the eigenvalues that determines the type and stability of the origin, as an isolated equilibrium of a nondegenerate linear system.  But it is the magnitude of the eigenvalues of a linear map from the plane to itself, that determines the type and stability of the isolated fixed point at the origin.  The secret as to why lies in calculating the time-1 map of a linear system of ODEs.  THAT is a linear map from the plane to the plane.  But that linear map does NOT have the same matrix as the linear system.  It is actually the matrix exponential of the matrix that determines the linear system, and that is sometimes a bit difficult to calculate.  We will...
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Lecture 13 - Rigid rotations of the circle

Today we move on from maps and flows in the plane and into maps and flows on the circle.  Here again, we start with a relatively simply construction to help set the stage for more complicated stuff later on;  Rigid circle rotations.  Really, they come in only two types, depending on whether the rotation is rational or not (but even this takes a bit of definition.)  But classifying circle rotations allows us to define what it means for an orbit to be dense in a space, and to define what it means for a point in the space to be recurrent with respect to a map.  This leads to an important theorem whose statement actually defines a property;  The Weyl Equidistribution Theorem.  There are two applications here that are quite interesting.  However, due to the timing of this tour, we will not cover them.  Instead, we use circle rotations to begin the discussion of circle flows and toral maps and flows, where things get a little more subtle and co...
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