At an early age we learn how to tackle the problem of solving linear equations for unknowns. What about solving nonlinear equations for unknowns? In general there is not much which can be said. One promising strategy is to start with a problem whose solution is known and perturb it. Consider an equation where should be thought of as the unknown and as a parameter. The mapping defined by is assumed smooth. Suppose that so that we have a solution for . It is helpful to consider the derivative of with respect to at the origin. If the linear map is invertible then we are in the situation of the implicit function theorem. The theorem says that there exists a smooth mapping from a neighbourhood of the zero in to such that . It is also (locally) unique. In other words the system of equations has a unique solution for any parameter value near zero.
What happens if is degenerate? This is where Lyapunov-Schmidt reduction comes in. Suppose for definiteness that the rank of is . Thus the kernel of is one-dimensional. We can do linear transformations independently in the copies of in the domain and range so as to simplify things. Let be the standard basis in a particular coordinate system. It can be arranged that is the span of and the range of is the span of . Now define a mapping from to by where is the projection onto the range of along the space spanned by . Things have now been set up so that the implicit function theorem can be applied to . It follows that there is a smooth mapping such that . In other words satisfy of the equations. It only remains to solve one equation which is given by . The advantage of this is that the dimensionality of the problem to be solved has been reduced drastically. The disadvantage is that the mapping is not known – we only know that it exists. At first sight it may be asked how this could possibly be useful. One way of going further is to use the fact that information about derivatives of at the origin can be used to give corresponding information on derivatives of at the origin. Under some circumstances this may be enough to show that the problem is equivalent to a simpler problem after a suitable diffeomorphism, giving qualitative information on the solution set. The last type of conclusion belongs to the field known as singularity theory.
My main source of information for the above account was the first chapter of the book ‘Singularities and groups in bifurcation theory’ by M. Golubitsky and D. Schaeffer. I did reformulate things to correspond to my own ideas of simplicity. In that book there is also a lot of more advanced material on Lyapunov-Schmidt reduction. In particular the space may be replaced by an infinite-dimensional Banach space in some applications. An example of this discussed in Chapter 8 of the book. This is the Hopf bifurcation which describes a way in which periodic solutions of a system of ODE can arise from a stationary solution as a parameter is varied. This is then applied in the Case study 2 immediately following that chapter to study the space-clamped Hodgkin-Huxley system mentioned in a previous post.