Tutorial: Introduction to bifurcation theory, part 2
Channel: Nils Berglund
Date: 24/09/2023

In this second part of our introduction to bifurcation theory, we look at the method called normal forms. That method aims at simplifying the nonlinear terms of a differential equation as much as possible, leaving only so-called resonant terms. Bifurcations of these simplified equations can be studied more easily. We discuss two examples: the Hopf, or Poincaré-Andronov-Hopf bifurcation, and the Bogdanov-Takens bifurcation near a point with a double zero eigenvalue. 

Intro: 0:00
Recap of part 1: 0:13
Normal forms: 8:18
Homological equation: 15:41
1D bifurcations: 25:34
Hopf bifurcation: 35:04
Double zero eigenvalue: 56:49
Bogdanov-Takens bifurcation: 1:05:46

Related lecture notes: 
https://arxiv.org/abs/math.HO/0111177  (chapter 3)
https://arxiv.org/abs/math.HO/0111178  (chapter 2)
Other tutorials: https://www.youtube.com/playlist?list=PLAZp3rbgWLo1ZKxqNzl04oaQhpbv_5mg9

#bifurcation #dynamical_system #normal_form