## Week 2

• 3. There goes the neighbourhood
• 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.

 Chapter 4 - Local stability Chapter 4 overheads A neighbourhood Stability matrix Re. 8:46 statement that the stability matrix [dv/dx] is not a Jacobian matrix, read Remark 4.1 Linear flows. Finite time linearized flow Jacobian matrix eigenvalues, eigenvectors Stability multipliers, discrete time Stability exponents, continuous time Complex multipliers, periods of rotation Jacobian transports its eigenframe Computing the Jacobian Jacobian as the time-ordered exponential Jacobian matrices form a semigroup Jacobian matrices in discrete time Jacobian matrix for a Poincare section

• 4. Cycle stability
• 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.

 Chapter 5 - Cycle stability Chapter 5 overheads Flow invariant sets Equilibria Periodic orbits

• Homework 2
•  Stability of equilibria and Floquet exponents Discussion forum for week 2

• Optional
•  Chapter 6 - Lyapunov exponents Norms Re. 1:40 to 2:16 - lecturer wonders off cammera. We'll fix that later :) 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. Let me know if you come across any problems. PowerPoint vs blackboard in teaching advanced mathematics? Week 1 was recorded in a black room with the instructor talking into a teleprompter over a PowerPoint presentation. 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 a studio. Professional Education technicians refused to do that, so we recorded and edited the lectures ourselves. Still, 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 the Piazza poll what works for you. You are the only judge of pedagogy that I trust, and I can teach online either way. Attracting Domain of a Dynamical System A Complex Dance by Sara Lapan