  Nonlinear two-point boundary value problems arise in numerous areas of
application. The existence and number of solutions for various cases has been
studied from a theoretical standpoint. These results generally rely upon growth
conditions of the nonlinearity. However, in general, one cannot forecast how
many solutions a boundary value problem may possess or even determine the
existence of a solution. In recent years numerical continuation methods have
been developed which permit the numerical approximation of all complex
solutions of systems of polynomial equations. In this paper, numerical
continuation methods are adapted to numerically calculate the solutions of
finite difference discretizations of nonlinear two-point boundary value
problems. The approach taken here is to perform a homotopy deformation to
successively refine discretizations. In this way additional new solutions on
finer meshes are obtained from solutions on coarser meshes. The complicating
issue which the complex polynomial system setting introduces is that the number
of solutions grows with the number of mesh points of the discretization. To
counter this, the use of filters to limit the number of paths to be followed at
each stage is considered.
