Bifurcation in the sense of applied mathematics happens when the system is on the boundary of one set of equivalent states. Generally a system undergoing bifurcation is in a critical state. Any small change in the parameters may result essentially different behaviors. This phenomenon is called the structural instability and is of great interest in (Lyapunov) stability investigations of engineering problems. In numerical simulations such property may cause several problems. We concentrate on the differences in the qualitative behavior of the solutions for recursive and anticipatory systems generated by the same mechanical model. The results show how the nature of solution depends on the type of bifurcation.