Index of a curve (with respect to a given vector field). Properties of the index. Index of a point. Using index theory to rule out closed trajectories. Some strange things: Index theory in biology. Hairy ball theorem. Combing a torus and connection to tokamaks and fusion. Index theory on compact orientable 2-manifolds.
Limit cycles. Examples in science and engineering. Definitions, and why linear systems can't have limit cycles.
Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 6.8, 7.0, 7.1.