Liapunov-Schmidt Method

Abstract

To study steady state and Hopf bifurcations, we introduce the well-known Liapunov-Schmidt method and derive an underlying low dimensional system of algebraic equations. This system is responsible for the bifurcation scenario and normally is easy to analyze. Another advantage of this approach is that the established singularity theory can be utilized directly to determine normal forms of these algebraic equations and their bifurcation scenario, see e.g. the monographes Golubitsky et al [129, 131] and Vanderbauwhede [295]. For numerical purposes Jepson et al generalize the Liapunov-Schmidt method in several aspects in a series of papers [173, 174, 175, 176]. We adapter in this chapter the discussion of Liapunov-Schmidt method in Ashwin/Böhmer/Mei [22] and a scaling technique in Mei/Schwarzer [226]. Center manifold reduction is another approach for analyzing local bifurcations and will be discussed in Chapter 7.