A unified analysis of reaction-diffusion equations and their finite difference representations is presented. Continuous and discrete problems are studied from the perspective of bifurcation theory. The numerical instability is shown to be associated with the bifurcation of periodic orbits in discrete systems. The new work presented in this paper is an analytical description of the nonlinear interaction of a high wavenumber mode, which is a product of the discretization, and a low wavenumber mode presented in the governing differential equation.