Analysis of system dynamics after saddle-node bifurcations for general nonlinear systems with unmodelled dynamics

The problem of losing the stability of equilibrium points of general nonlinear autonomous dynamical systems due to a single parameter associated with the saddle-node bifurcations is considered. The cases when some of the state variables are assumed to be fixed at the bifurcation or under the effect of small changes are analyzed. It is shown analytically that under some conditions saddle-node bifurcations are persistent with regular or singular perturbations of the vector field. It is also shown that the dynamics after the bifurcation can be identified and easily described