Bifurcation in Difference Approximations to Two-Point Boundary Value Problems

Numerical methods for bifurcation problems of the form (*) Ly = Xf(y), By = 0, where/(0) = 0 and /'(0) ^ 0, are considered. Here y is a scalar function, A. is a real scalar, L is a linear differential operator and By = 0 represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if (*) is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of (*) bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.