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

Explore hyperbolic and non-hyperbolic fixed points in dynamical systems and learn techniques for computing their invariant manifolds using Taylor series expansions. Delve into the stable, unstable, and center subspaces for discrete-time dynamical systems, and understand the significance of corresponding invariant manifolds in nonlinear settings. Examine a 2D example of analytically obtaining stable and unstable manifolds, and discover methods for approximating invariant manifolds. This lecture, part of a course on center manifolds, normal forms, and bifurcations, provides essential insights for students with a background in elementary analysis, multivariable calculus, and linear algebra.

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.