The Degenerate Node

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.