Minimal complexity of semi-bounded components in bifurcation theory Original Research Article

In this paper we use the topological degree to estimate the minimal number of solutions of the λ-slices of the semi-bounded and bounded components of a general class of fixed point equations depending on a real parameter λ by means of the signature of the component. The signature is defined as the set of bifurcation intervals from a given state together with all changes of the local index of the state. Calculating the minimal number of solutions of the component slices enables us to ascertain its minimal complexity in terms of graph theory, at least for a generic family of non-degenerate situations, where the general theory of this paper provides the minimal (canonical) structure of the component.