What is Bifurcation Theory?

In the following I would like to present the main ideas of bifurcation theory along with some basic examples that illustrate the theory. For the sake of shortness let me formulate the fundamental question of bifurcation theory in an abstract way. Suppose you are given an equation \(F(x,\lambda)=0\) with the property that \(x=0\) is always a solution, the so-called trivial solution. Is it possible to find some \(\lambda^*\) such that a sequence of nontrivial solutions converges to \((0,\lambda^*)\)? In other words: Do nontrivial solutions bifurcate from the trivial solutions? In the following I will present three equations from analysis, linear algebra and ordinary differential equations showing that bifurcation theory is a topic worth studying! The reading requires some amount of advanced mathematics — do not hesitate to contact me if you need some additional explanations. By the way: Next semester I will give a lecture on that topic which is suited for master students or advanced bachelor students with a background in analysis, differential equations and possibly boundary value problems, see below for more information. Continue Reading →