Abstract: Bifurcations with constraints are open problems appeared in research on periodic bifurcations of nonlinear dynamical systems, but the present singularity theory doesn’t contain any analytical methods and results about it. As the complement to singularity theory and the first step to study on constrained bifurcations, here are given the transition sets and persistent perturbed bifurcation diagrams of 10 elementary bifurcation of codimension no more than three.

Key words: constraint, singularity theory, bifurcation, numerical wave tank, free surface simulation, Navier-Stokes equations, staggered grids, clustering technique, wave generation