Abstract
A time-dependent bifurcation model and its control problem are studied. Firstly, delayed bifurcating transition phenomena with memory effects of the model with time-dependent parameters varying in either the positive or the negative direction are analysed. Secondly, a parametric control problem with feedback for the time-dependent model is investigated. The existence and stability of dynamical hysteresis cycles are obtained by qualitative analysis of bifurcation and stability. Finally, an important mechanism for dynamical hysteresis and pulsing oscillation in parametric control systems is revealed as the result of delayed bifurcating transitions when the bifurcation parameter varies periodically across the steady bifurcation value.