A PDF copy of the syllabus can be found here: APMA_2190.pdf

December 5: We discussed the Lyapunov–Schmidt reduction and proved the Crandall−Rabinowitz theorem on bifurcations from simple eigenvalues. The original paper by Crandall and Rabinowitz can be found here:

Michael Crandall and Paul Rabinowitz, Bifurcation from simple eigenvalues Journal of Functional Analysis, 8(2), 321-340, 1971.

November 28: We proved the implicit function theorem in Banach spaces and focused on its usage in local bifurcation theory.

The fourth homework assignment has been posted: HW4.pdf

November 16: We discussed the notions of Fréchet and Gâteaux differentiability, and we proved the inverse function theorem in Banach spaces.

November 14: We proved some basic relations between the value of the index of a closed path and the existence of critical points for a planar dynamical system. In this context, we also discussed the notion of the index of an isolated critical point.

November 9: We introduced the notion of the index of a closed path with respect to a vector field in the plane and proved its invariance under homotopies.

November 7: We completed the proof of the Poincaré-Bendixson theorem and characterized all possible ω-limit sets of a planar flow in a positively invariant domain with finitely many rest points.

October 31: We discussed some basic results concerning transversality and ω-limit sets for planar flows, and we proved the Poincaré-Bendixson theorem (to be completed in the next lecture).

October 26: We proved the Hadamard-Perron theorem on the existence and regularity of invariant manifolds (see also problem 5 in assignment 3).

The third homework assignment has been posted: HW3.pdf

October 24: We defined the invariant manifolds of a hyperbolic critical point and discussed the equivalence of the existence of invariant manifolds to the existence of fixed points for certain properly defined operators.

October 17: We introduced the notion of a fiber contraction and proved the fiber contraction theorem. This will be used in the proof of the Hadamard-Perron theorem in subsequent lectures.

October 12: We completed the proof of the Grobman-Hartman theorem on the topological classification of the dynamics in a neighborhood of a hyperbolic critical point. Philip Hartman's original paper on this topic can be found here

October 10: We used the contraction mapping theorem to establish the existence of topological conjugacies for small Lipschitz perturbations of hyperbolic linear flows.

October 3: We laid the groundwork for proving the Grobman-Hartman theorem and introduced the notions of hyperbolicity and topological conjugacy.

The second homework assignment has been posted: HW2.pdf A nice discussion of some of the consequences of problem 4 can be found in this article by Arieh Iserles.

September 28: We discussed some basic elements of Floquet theory. We proved Floquet's theorem and identified conditions on the monodromy matrix under which a linear system with periodic coefficients has periodic solutions.

September 21: We focused on the results pertaining to extensions of local solutions and continuous dependence on initial conditions. We introduced the notion of a fundamental solution of the system $x^{'} = A (t) x$ and derived Liouville's formula.

September 19: We discussed two different proofs of Peano's theorem; one that relied on approximating the continuous vector field by a family of Lipschitz vector fields and using the Picard-Lindelöf theorem, and one that was based on a rather more explicit construction of the approximating sequence. The core idea behind both proofs was a compactness argument based on the Arzelà-Ascoli theorem.

September 14: We proved the Arzelà-Ascoli theorem, which provides a characterization of the compact subsets of $C\left(\left[a,b\right],\mathbb{R}^n\right)$ , and we discussed an approximation method for deriving the proof of Peano's theorem (to be completed in the next lecture).

The first homework assignment has been posted: HW1.pdf

September 12: We proved that $C\left(\left[a,b\right],\mathbb{R}^n\right)$ equipped with the sup norm is a Banach space, and we used this result in conjunction with the contraction mapping theorem to derive the proof of the Picard-Lindelöf theorem.

September 7: We reviewed some basic material about complete metric spaces and proved the contraction mapping theorem (see, e.g., theorem 1.35 in the text by Barreira and Valls).