I. Existence and Structure of Bifurcation Branches
The problem of bifurcation is formulated as an operator equationin a Banach space, depending on relevant control parameters, say ofthe form G(u,λ) = 0. If dimN(G_u(u_O,λ_O)) = m the method of Lyapunov-Schmidtreduces the problem to the solution of m algebraic equations.The possible structure of these equations and the various types ofsolution behaviour are discussed. The equations are normally derivedunder the assumption that G^O_λεR(G^O_u). It is shown, however, thatif G^O_λεR(G^O_u) then bifurcation still may occur and the local structureof such branches is determined. A new and compact proof of theexistence of multiple bifurcation is derived. The linearizedstability near simple bifurcation and "normal" limit points is thenindicated.
II. Constructive Techniques for the Generation of Solution Branches
A method is described in which the dependence of the solutionarc on a naturally occurring parameter is replaced by the dependenceon a form of pseudo-arclength. This results in continuation proceduresthrough regular and "normal" limit points. In the neighborhoodof bifurcation points, however, the associated linear operatoris nearly singular causing difficulty in the convergence of continuationmethods. A study of the approach to singularity of thisoperator yields convergence proofs for an iterative method for determining the solution arc in the neighborhood of a simple bifurcationpoint. As a result of these considerations, a new constructiveproof of bifurcation is determined.
