A question of central importance in the numerical approximation of time evolving equations (reaction-diffusion-convection equations) is whether or not the simulation produces the same asymptotic behaviour as the underlying continuous problem, for fixed but small values of mesh- spacings. This question is closely related to the existence of spurious steady, periodic and quasi-periodic solutions generated by discretizations: linear instability (where the linearization is about any steady solution of the difference equations) implies the existence of spurious periodic solutions in the fully nonlinear problem.
A profound and fairly comprehensive analysis gives an answer to the question whether or not the periodic solutions can exist for arbitrarily small values of the time-step and what form they take. The method is a combination of local bifurcation theory and the study of modified singularly perturbed boundary value problems by singular perturbation techniques. Especially, a local construction, via bifurcation theory, of the spurious periodic solutions near to the critical values of time steps at which they bifurcate from the steady solution, is found. - Again, a thorough, exhausted study.