Pitchfork and Fold/Fold bursting of time delay duffing systems with parametric excitation and external excitation

Time delay is very common and inevitable in engineering. In this paper, a time delay duffing system with combining parametric and external excitation has been analyzed. First, the system with only parametric excitation shows pitchfork bifurcation. When the slow variable experiences the pitchfork bifurcation, the system will not jump immediately but moves along the unstable bifurcation for a while, and then shows the bursting oscillation. Time delay makes the system randomly choose to go up or down into the bursting oscillation, which may be due to the influence of different regions of attraction. Second, the system with parametric and external excitation shows Fold/Fold bursting. We can construct different paths to make the system bursting including linear, vertical parabolic and horizontal parabolic. At the same time, we find that the phase difference between parametric excitation and external excitation has a vital effect on the realization of bursting oscillation. Finally, we analyze that the positive time delay will not occur Hopf bifurcation, therefore the dynamical behavior of systems with time delay is similar to that without time delay.