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
Classification of equilibrium points in the plane with examples (e.g. Verhulst Section 3.1 or Teschl, Section 3.2 - in particular Figures 3.1-3.5)
Discussion of Exercise Sheet 2; Lyapunov stability vs. asymptotic stability
Gronwall lemma (presented in the form of Verhulst, Theorem 1.2; Teschl, Lemma 2.7 gives a more general version; what we actually need and use is even more special than the version of Verhulst); use of the Gronwall lemma to prove a theorem on the asymptotic stability of the trivial solution (simplified version of Verhulst’s presentation of the Poincaré-Lyapunov theorem, Theorem 7.1; Teschl states a basically equivalent version as Theorem 6.1 but the proof is scattered through the book. See, however, the start of Teschl, Section 6.5 for basic definitions and examples)
Discussion of Exercise Sheet 3
Periodic solutions, Bendixon’s criterion, limit points, omega-limitsets, Poincaré-Bendixon theorem without proof (selected topics of Verhulst, Chapter 4, or Teschl, Section 6.3 up to Lemma 6.7, Theorem 7.16 without proof; Bendixon’s criterion is basically Teschl, Problem 7.11)
Discussion of Exercise Sheet 4, here, in particular, how to use Sympy to perform the stability analysis required in Exercise 4
Discussion of Exercise 4 (ctd.)
Discussion of Exercise 4 (ctd.)
Stability via Lyapunov functions (following Teschl, Section 6.6)
Examples for stability analysis via Lyapunov’s method: Hamiltonian systems, gradient flows, ad-hoc construction of a Lyapunov function (for Hamiltonian systems and gradient flows, the basic statements are in Teschl, 6.6 and 6.7, presentation from a slightly more abstract point of view; the final example is from Verhulst, Section 8.1, but also presented in the language of the Lyapunov theorem)
Example for Hamiltonian system: mathematical pendulum (parts of Teschl, Section 6.7); Elementary bifurcations of one-dimensional vector fields: saddle-node, transcritical, pitchfork
Discussion of Exercise 5; bifurcations and the implicit function theorem
Hopf bifurcation
Discussion of Exercise 5
Center manifolds