Bifurcation in difference approximations to two-point boundary value problems

Abstract:

Numerical methods for bifurcation problems of the form \begin{equation}\tag {$\ast $} Ly = \lambda f(y),\quad By = 0,\end{equation} where $f(0) = 0$ and $f’(0) \ne 0$, are considered. Here y is a scalar function, $\lambda$ 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 $(\ast )$ 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 $(\ast )$ 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.