Hyperbolic vs Non-Hyperbolic Fixed Points - Computing Their Invariant Manifolds via Taylor Series

This lecture examines fixed points of discrete-time maps, distinguishing linear stable, unstable, and center subspaces from nonlinear invariant manifolds. It includes a two-dimensional analytical example and introduces Taylor-series approximations, with prerequisites in elementary analysis, multivariable calculus, and linear algebra.

Syllabus

Fixed points of maps and their stable, unstable, and center subspaces.
Subspaces (linear) vs. invariant manifolds (nonlinear).
Hyperbolic vs. non-hyperbolic fixed points .
Diagram of hyperbolic vs. non-hyperbolic fixed points .
Why look at center manifold theory?.
2D example of calculating an invariant manifold analytically.
Approximating invariant manifolds via Taylor series expansion.