Methods of Representation Theory and Group Analysis in Bifurcation Theory

The application of group representation methods in branching theory began with the pioneering works of V.I. Yudovich both in stationary branching [1–3] and non-stationary branching [4, 5]. The historical material of this subject is contained in the introduction. In this chapter the survey of investigations on the usage of continuous and discrete group symmetry for the construction and investigation of Lyapounov—Schmidt branching equations (BEq) at stationary and dynamic bifurcation is given. In this direction the differential equations group analysis methods based on the theorem about BEq symmetry inheritance and S.Lie—L.V.Ovsyannikov [1,2] theory of invariant manifolds turn out to be more effective. The notion of resolving systems for bifurcation theory problems is introduced, in particular, for differential equations in Banach spaces with a degenerate operator at the derivative. It allows one to investigate bifurcating solutions stability questions.