Introduction to bifurcation theory

The theory of bifuxcation from equilibria based on center-manifold reduction and Poincare-Birkhoff normal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and recent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest generic bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation occurring in a model of thxee-wave xnode coupling and steady-state bifurcations occurring in the real Landau-Ginzburg equation. The forxner provides an example of the importance of degenerate bifurcations in problexns with more than one parameter and the latter illustxates new effects introduced into a bifurcation problexn by a continuous symmetry.