Week 2

TEXT   Week 2 Overview

  • 3. There goes the neighbourhood / 13 January 2015 /
  • So far we have concentrated on description of the trajectory of a single initial point. Our next task is to define and determine the size of a neighbourhood, and describe the local geometry of the neighbourhood by studying the linearized flow. What matters are the expanding directions. The repercussion are far-reaching: As long as the number of unstable directions is finite, the same theory applies to finite-dimensional ODEs, Hamiltonian flows, and dissipative, volume contracting infinite-dimensional PDEs.

    Reading: Chapter 4   Chapter 4 - Local stability
    Overheads: Chapter 4   Chapter 4 overheads
    Video lecture: Stability1neighb.mp4   A neighbourhood
    Video lecture: Stability1stabMat.mp4   Stability matrix
    Re. 8:46 statement that the stability matrix [dv/dx] is not a Jacobian matrix, read Remark 4.1 Linear flows.
    Video lecture: Stability1finTime   Finite time linearized flow
    Video lecture: Stability1jacob   Jacobian matrix eigenvalues, eigenvectors
    Video lecture: ?   Stability multipliers, discrete time
    Video lecture: Stability1jacobContT   Stability exponents, continuous time
    Video lecture: Stability1jacobFreq   Complex multipliers, periods of rotation
    Video lecture: Stability1eigenvect   Jacobian transports its eigenframe
    Video lecture: Stability1compJac   Computing the Jacobian
    Video lecture: Stability3timeOrd   Jacobian as the time-ordered exponential
    Video lecture: Stability3semiGroup   Jacobian matrices form a semigroup
    Video lecture: Stability3jacobDiscrT   Jacobian matrices in discrete time
    Video lecture: Stability3jacobPoinc   Jacobian matrix for a Poincare section

  • 4. Cycle stability / 15 January 2015 /
  • If a flow is smooth, in a sufficiently small neighbourhood it is essentially linear. Hence in this lecture, which might seem an embarrassment (what is a lecture on linear flows doing in a book on nonlinear dynamics?), offers a firm stepping stone on the way to understanding nonlinear flows. Linear charts are the key tool of differential geometry, general relativity, etc, so we are in good company. If you know your eigenvalues and eigenvectors, you may prefer to fast forward here.

    Reading: Chapter 5   Chapter 5 - Cycle stability
    Overheads: Chapter 5   Chapter 5 overheads
    Video lecture: Flow invariant sets invariants1sets   Flow invariant sets
    Video lecture: Equilibria invariants1equi   Equilibria
    Video lecture: Periodic orbits invariants1POs   Periodic orbits

  • Homework 2
  • Homework   Stability of equilibria and Floquet exponents
     Due 27 January 2015
    Homework   Discussion forum for week 2

  • Optional
  • Reading: Chapter 6   Chapter 6 - Lyapunov exponents
    Video lecture: invariants1norms   Norms
    Re. 1:40 to 2:16 - lecturer wonders off cammera. We'll fix that later :)
    Video lecture: discreteD1   Double Double-pendulum Android app

    An app from our instructor Ashley:

    I had to do a public talk and needed a demonstration piece for the double pendulum, which I (incorrectly) thought might be easiest to do with the android/opengl libraries. I thought it would be easier to track periodic orbits in the system than it turned out to be. I've added a trace to the pendulum while showing the nearest periodic orbit, which makes it easier to see examples of shadowing. Let me know if you come across any problems.

    Video lecture: discreteD1   To FeeblePoint or not to FeeblePoint

    Now you have seen what the administrators love:

    Week 1 was a talking head in a black room blabbing into a teleprompter over a "Power"Point presentation, while the video technician is getting a coffee in the next room. We (professors teaching advanced mathematics) have tried this in teaching a decade ago, concluded that one cannot learn this way, and went back to live teaching on blackboards, with multicolored chalks, in real time.

    Week 2 is what makes this course "experimental". I expect you to study a Guttenberg-kind of text, and sweat over every step of a derivation, until you understand the result and understand what it means. Skipping ahead will get you nowhere. Either you learn this stuff, or we are both wasting our time. You click on a video link, maybe, to relax or to see how your teacher thinks about the problem.

    The experiment is labour intensive: a video technician would have to record lectures in my classroom, live, not let the camera run in "Professional" "Education" black room. They refused, so we recorded the lectures ourselves. They are into "branding". They demanded that the Georgia Tech PP slide be tacked onto every video, so I gave them the videos. The course was nearly stopped in October, as they insisted that the branding Georgia Tech PP slide have my photograph face-on, while I used a profile photo of me in front of (gasp!) blackboard. You, the feeble minded student, might forget between two slides that this course is brought to you by Georgia Tech, with a certificate signed by an administrator who has no clue what the course is about.

    A junior video technician refused to edit them (trust me - it is work; I do it), and told the administrators to stop the course. So they did, without any warning or a prior communication to the instructor. The simple, lame truth. We never asked for "help" and we never got any, except for what is acknowledged on the bottom of Week 1 homepage.

    Now, I personally do not know these administrators, I do not work for them. My understanding is that they work for me. All I can surmise about the administrators is that they draw administrative salaries, and as is the practice in administrations, bear no consequences for the "decisions" they make. They have never read the experimental course proposal that a Georgia Tech faculty committee funded, as the course is executed exactly as proposed. They did not spend their winter holidays reinventing the wheel of online education in three weeks, as Burak and I suddenly had to do. All they did is they signed Georgia Tech contract with the commercial company Coursera.org: we, the faculty, provide the content, for free, for the love of teaching and love of learning. The administrators were never put in charge of our graduate curriculum. The administrators have never taught an advanced mathematical physics course. I, with some 40 years plus of teaching graduate courses, with all my shortcomings as an educator, have developed some sense of what works in teaching.

    Still, the world is rapidly changing, we might be at crossroads, and I do not really know whether what I do helps you learn. I would love to hear what is better for YOU: 5 minute Power Point videos, or a slow study of a book (enhanced by some live lecture video hyperlinks, but still a book). Let us know in a Piazza forum what works for you. You are the only judge of pedagogy that I trust, and I can teach online either way.