Course work and assessment:
There is regular homework in which students use analysis and/or simulation to apply the ideas of the course. There is a course project in which students apply the course methods to a problem in their field of interest or study a course topic in depth. Students are assessed on their performance in the homework and the project.

Course topics in detail (may vary):
Motivation and comments are enclosed in square brackets [].
Generally theorems are carefully stated but not proved.

- Stability, periodic orbits, and chaos in logistic, tent and binary shift maps. Use of implicit function theorem to show robustness of hyperbolic fixed points with respect to parameter variations. [illustrate main ideas of course in simpler one dimensional map context.]

- Statement of differential equation existence, uniqueness and smoothness of solutions. Exponential separation of solutions for Lipschitz vector fields. Statement and explanation of implicit function theorem. Definitions of hyperbolicity and topological equivalence. [Review basic tools and set up notation]

- Manifolds, tangent planes and transversal intersections. [Crash intro to basic differential geometry]

- Hartman Grobman theorem, tubular flow [i.e. What does a vector field look like near and far from equilibrium?]

- Linear system phase portraits and invariant hyperplanes [eigenspaces] [review of linear systems with new language] Preview of suspension and Poincare map in the case of the sinusoidally forced pendulum.

- Stable and unstable manifolds for a hyperbolic equilibrium: definition, local construction by Hartman Grobman, global construction by flow. Statement of stable manifold theorem [to get smoothness of stable and unstable manifolds].

- Structural stability.

- Stretching and compressing action of linear flows on sets of initial conditions near 3 dimensional saddles. [This is a key mechanism underlying chaos in the pendulum]

- Poincare map: definition, smoothness of return time and Poincare map, preservation of orientation. Correspondence between periodic orbits and Poincare map fixed points.

- Bifurcation concept. Standard one dimensional examples of saddle nodes of flows and maps. Generic saddle node in one dimension; transversality conditions lead to robustness of the bifurcation. Discussion of generic = open+dense. [This material links mathematical genericity to robustness of models].

- Saddle node in many dimensions, Lyapunov-Schmidt reduction.

- Pitchfork bifurcation, odd symmetry and genericity. Period doubling bifurcation of maps

- Cantor sets as invariant sets of tent map dynamics; associated symbolic dynamics.

- Haussdorf metric; ideal fractals as attracting invariant sets of sets of contractions. Rossler folded band attractor.

- Smale horseshoe and its symbolic dynamics.

- Detection of chaos in the sinusoidally forced pendulum with Melnikov's method.

- Lambda lemma; horseshoes in the pendulum.

- Center manifolds; lack of smoothness and uniqueness, computing approximations.

- Normal forms; normal form for Hopf bifurcation. Hopf theorem.

- Fractal basin boundaries (stable manifold immersion) in the forced pendulum.

- Takens embedding theorem, delay and embedding dimension, reconstruction of attractors from time series, dripping faucets. [This material is of particular use to experimentalists]