The Degenerate Node

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.