System delay in a haptic virtual world creates inappropriate instability and oscillations. An intrinsically unstable environment can give appropriate instabilities such as snap buckling. Around a buckling point the low-order approximations useful in solution methods become degenerate, overlaying a non-delay oscillation on the jumps required by physical realism. We show how bifurcation theory can identify and limit the higher terms needed for a correct solution by algorithms with haptically viable speed. We illustrate this in a haptic simulation of stiff thread bending as the hand moves it, with realistic 'snap' buckling, and in the first haptic simulation of the non-linear Zeeman machine, a canonical system for the study of energy-minimizing jumps amid continuity.