The class Differential Equation will focus this semester on dynamical systems techniques in the theory of differential equations. We will cover local existence and uniqueness, systems of linear equations, stability of equilibrium points of nonlinear equations, bifurcations, the Poincaré-Bendixon theorem, chaos, and, if time allows, a selection of perturbation techniques and data-driven methods.
There are several very good modern textbooks on the topic. I will base the general structure of the course roughly on parts of the book by Verhulst, but link to chapters in Teschl’s book as much as possible, as this is the only book that appears to be legally and freely available.
Introduction, contraction mapping theorem (see, e.g., Teschl, Section 2.1)
Fundamental existence and uniqueness theorem (e.g. Teschl, Section 2.2), basic examples of the type \(\dot x = x^\alpha\) for different values of \(\alpha\)
Solutions for scalar first-order equations via an integrating factor, separable equations (e.g. Teschl, Section 1.3 and start of Section 1.4, but with different examples); energy estimate to prove continuous dependence on the initial data
Autonomous equations: equilibrium points and linearization; Solution of linear constant-coefficient equations via the matrix exponential (e.g. Teschl, Section 3.1)
Discussion of Exercise Sheet 1