The Degenerate Node

One fun application of phase volume preservation in a dynamical system is billiards.  The polygonal table introduced recently when developing toral flows is an example of a general class of dynamical systems called convex billiards.  Today, we will describe the system and talk about its properties.  One interesting aspect of a convex billiard (essentially the table is a convex domain in the plane) is that it always has periodic orbits of all orders.  This may not be obvious, but we will define the tools that expose why.  And in one particular case, that of an elliptic billiard,  there is a curious side effect of the table;  given a particular placement of a pocket and ball placement, one can aim anywhere and cannot miss making the shot.  Makes for good betting odds, eh? 

Now we will leave circle and toral maps for a bit and change our focus a bit.  We will stay on this notion of recurrence, however, but in a different fashion.  Instead of simply looking for a particular orbit to exhibit certain behavior, let's look at families of orbits simultaneously.  In particular, for a flow or a map, let's look at the orbit of a small open set in state (phase space), as in consider all points in a small open set to be the initial points and follow all of their orbits at once.  Today we will introduce the notion of incompressability of a flow, which is also called phase volume preservation.  All of our toral translations and flows to date preserved volume, as do all isonetries, but contractions do not.  For a flow, this is related to divergence of the vector field of the flow, and we show that Newton;s Second Law of Motion provides a rich set of examples, along with exact differential equations.  Then we show that the ramifications of volume preservatio

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.

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 complicated. 

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 spend a short time on this. 

For today, we will discuss linear maps of the plane, leading to a classification that is quite similar to that of linear systems of ODEs, but with some important differences.  Once classified, we will use a particular example to highlight what divergence may mean for an orbit of a linear dynamical system in real two space and just how an orbit can go to infinity.  From this, we get a way to create a functional form for the elements of a Fibonacci (or similar) sequence. 

So for today, we will continue to take some time to introduce and study some of the objects of dynamics that go beyond maps and ODEs.  Mainly, some common spaces that act as domains for maps, or the places that serve as playgrounds for orbits and trajectories.  It turns out that the properties of a dynamical system do not only rely on the properties of a map, but also on the properties of the domain of a map. Once we define a couple more things like isometries and equivalence of metrics, we will define a few spaces that we will study more closely later, like spheres and tori.  We will spend some time on the simple space of a circle, since defining it will be important to understanding it.  And we will also define a new space, useful in dynamics and weirdly mesmerizing in analysis in general;  The Cantor Set.   Most likely,m we will end here. See you today.

So for tomorrow, I will leave our final example of a 3-dimensional Poincare Map (a First Return Map) and discuss just one more example of relatively simply dynamics, that of the Logistic Map (Section 2.6).  It is a family of maps, parameterized by a single parameter.  It is also a very complicated one.  However, for a certain interval of parameter values, it is quite simple to describe the dynamics.  Although even here, there are clues to future complications.  We will return to the example I ended with at a later date. After we spend some time with the Logistic Map, we will step through some of the parts of Chapter 3 that will be important for later. Chapter 3, in the whole, is a place to dig deeper into some of the more subtle aspects of spaces, metrics and continuous maps that are the objects of dynamical systems.  We will touch on a few topics, like equivalent metrics, continuity using metrics, and some non-Euclidean spaces.  We will leave the rest for moments when context di

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.