Bifurcations in One Dimension, and Flows on the Circle
Stub. How a 1D flow's qualitative structure changes as a parameter varies — the saddle-node, transcritical, and pitchfork bifurcations, with laser threshold and insect-outbreak applications and imperfect bifurcations/catastrophes — then flows on the circle, the next-simplest phase space, where (unlike the line) oscillation becomes possible: uniform and nonuniform oscillators, the overdamped pendulum, fireflies, and Josephson junctions.
Corresponds to Strogatz, Chapters 3–4. A bifurcation is a qualitative change in a flow’s fixed-point structure as a parameter crosses a critical value — fixed points appearing/disappearing in pairs (saddle-node), exchanging stability (transcritical), or splitting symmetrically (pitchfork) — worked out first in the 1D setting of the previous note, with physical examples (a laser turning on, an insect population exploding) and the complication of imperfect (symmetry-broken) bifurcations and catastrophe-theory jumps. Flows on the circle keep the local 1D-flow analysis but change the global topology just enough to allow genuine periodic behavior — the simplest model of an oscillator — illustrated with synchronizing fireflies and Josephson junctions.
Outline (TODO — flesh out each)
- Saddle-node bifurcation
- Transcritical bifurcation
- Laser threshold
- Pitchfork bifurcation
- Overdamped bead on a rotating hoop
- Imperfect bifurcations and catastrophes
- Insect outbreak
- Flows on the circle — examples and definitions
- Uniform and nonuniform oscillators
- Overdamped pendulum
- Fireflies; superconducting Josephson junctions