1) Prerequisites
Assumed is knowledge of ordinary differential equations as taught in bachelor courses.
Such knowledge is covered for instance in "An Introduction to
Dynamical Systems" by R. Clark Robinson, in Chapters 1-6 (that treat existence and uniqueness, linear systems, concept of a flow, limit sets and stability, periodic orbits).
2) Aim of the course
Bifurcation theory is the mathematical study of changes in the structure of solutions of a dynamical system under variation of parameters.
This course provides a thorough overview of bifurcation theory, mainly in the context of differential equations. It gives a detailed discussion of the main local and nonlocal bifurcations and also discusses global bifurcation phenomena, such as routes to chaos given by intermittency or period-doubling cascades. It develops a broad range of techniques, both geometric and analytic, for studying bifurcations. Techniques include normal form methods, center manifold reductions, the Lyapunov-Schmidt construction and Melnikov's method. The importance of the theory is made clear using carefully chosen applications.
3) Rules about Homework/Exam
The grade will be composed of a written exam and homework sets.
The homework sets count for 30% of the grade. A minimum grade of 5.0 for the written exam is required to pass. The homework does not count as part of the grade after retake.
4) Lecture notes/Literature
A.J. Homburg, J. Knobloch. Bifurcation Theory. Grad. Stud. Math. 246. American Mathematical Society, Providence, RI, 2024.
- Docent: Arjen Doelman
- Docent: Ale Jan Homburg