Title:

The application of Newton's method to simple bifurcation and turning point problems

This thesis investigates the solution of equations of the form G(Lambda,chi)=0, G: R chi X > X, where X is a real Banach space. Such equations are often called nonlinear eigenvalue problems. If (Lambda,chi) is a solution for which Gx(Lambda,chi) is invertible, there are wellknown existence and uniqueness results for solutions near (Lambda,chi), which are easily made constructive. However, in this thesis, we are interested in solutions (Lambda,chi) for which Gx(Lambda,chi) is not invertible, and specifically in the case when Gx(Lambda,chi) has only a ldimensional nullspace. Our approach is to apply the NewtonKantorovich theorem, first to determine these socalled singular points, and second to compute nearby solutions. In the former case we modify the equations to avoid singular systems, and in the latter case we obtain accurate starting values which compensate for the nearsingularity of Gx(Lambda,chi). Hence the advantages of a quadratically convergent method are retained. Chapter 1 contains a brief, general introduction to nonlinear eigenvalue problems in which we distinguish between the two important types of singular point, turning points and bifurcation points. In Chapter 2 we consider the common case G(Lambda,O) = O for arbitrary Lambda. Thus (Lambda,o) is a solution, the singular points (Lambda,O) can be computed by standard methods, and the major difficulty is to determine solutions (Lambda,chi), near (Lambda,O), with nonzero chicomponent. Turning points are the subject of Chapter 3, where we show how to compute both the points themselves and the solutions beyond them. In Chapter 4 we consider the problem of bifurcation points in a similar way and, finally, the stability of such points under small perturbations is discussed in Chapter 5.
