Phase-Space Geometry, Fixed Points, and Linearization
Stub. The basic vocabulary for studying a dynamical system geometrically rather than by closed-form solution: states and evolution rules, flows versus maps, the phase-space picture, fixed points, and linearization/stability via the Jacobian — including the discrete-time (repeated-Jacobian) case needed for recurrent neural networks.
Corresponds to lectures: Course introduction (8/25); Phase-space geometry and fixed points (8/27); Linearization and stability (9/1); Maps, timesteps, and repeated Jacobians (9/3) — the first quarter of the syllabus’s “Foundations and computational tools” unit (weeks 1–4). Sets up the same geometric phase-space viewpoint as introduction-and-flows-on-the-line and linear-systems-and-the-phase-plane, but aimed at a different application: understanding trained and untrained neural networks (which are themselves high-dimensional dynamical systems, continuous-time flows for some architectures and discrete-time maps for others, like an RNN applying the same update repeatedly) rather than classical physical systems. The key technical object is the Jacobian at a fixed point — for a continuous flow it gives the usual eigenvalue stability picture, but for a discrete-time map applied times, stability depends on the product of Jacobians, which need not even commute — the complication that motivates tangent dynamics and Lyapunov exponents in the next note.
Outline (TODO — flesh out each)
- Course introduction — states and evolution rules
- Flows versus maps
- Phase-space geometry
- Fixed points
- Linearization and stability (continuous-time)
- Maps and timesteps — the discrete-time case
- Repeated Jacobians — stability of a map iterated many times