Persistence of saddle-node bifurcations for general nonlinear systems under unmodeled dynamics and applications

The authors consider a general nonlinear dynamical system with both slow and fast unmodeled dynamics called the original system. Associated with the original system, they define a simplified system which treats the fast variables in the original system as instantaneous variables and the slow variables as constants. It is shown that under a fair general condition, the general nonlinear system with both fast and slow dynamics will encounter a saddle-node bifurcation relative to a varying parameter if the associated simplified system encounters a saddle-node bifurcation relative to the varying parameter. An error bound is derived between the bifurcation point of the simplified system and that of the original system. It is shown that the system behaviors, after the saddle-node bifurcation of the reduced system and that of the original system, are close to each other in state space