Bifurcation theory allows us to systematically study the non-uniqueness of solutions to nonlinear problems in analysis. I will offer an accessible introduction to this subject, aimed at PhD students and with an emphasis on motivating examples.

After recalling the implicit function theorem, we will begin with systems of algebraic equations in finite dimensions, using Taylor expansions to motivate the classical Crandall–Rabinowitz local bifurcation theorem, and examining the so-called “transversality condition” from several different points of view. Next we will move to infinite dimensions, where our primary example will be a nonlinear elliptic partial differential equation. Formal calculations using separation of variables suggest a certain bifurcation diagram, and we will discuss the additional steps needed to make such an argument rigorous. Finally, we will turn to global bifurcation, continuing the above branches of solutions beyond the perturbative regime. Here we will present two results, a topological theory for smooth maps and an alternative theory for real-analytic maps, the first of which will be motivated by a rough “proof by picture” in two dimensions.