If during a physical experiment one parameter is slightly varied one expects in general no dramatic change but that the observed stationary physical states vary continuously with the parameter in a unique way. This is precisely what the Implicit Function Theorem predicts in a corresponding mathematical model. However, in some cases something unexpected happens if the parameter exceeds a critical threshold. Physicists call it a “self-organization of a new state”, a “spontaneous symmetry breaking”, an “onset of auto-oscillations” and so on. Mathematically spoken a bifurcation takes place, either a bifurcation of equilibria or of periodic solutions, the latter also called a Hopf Bifurcation. (Of course more complex solutions may emerge in some cases, too.) The new state is actually observed if there is an exchange of stability, which means that the ground state loses and the bifurcating state gains stability. Many striking examples of these phenomena can be observed in reality.

In this talk we give an overview of important bifurcation theorems for equilibria and explain these by examples and counterexamples. Finally, we discuss a specific example given by the Euler-Lagrange equation for the Cahn-Hilliard energy of a binary alloy.