Cluster Oscillation of a Fractional-Order Duffing System with Slow Variable Parameter Excitation

The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation are discussed in detail, and it was found that Hopf bifurcation only happens while the fractional order is bigger than 1. On the other hand, the influence of the amplitude of parametric excitation on cluster oscillation models is discussed. The results show that amplitude regulates cluster oscillation models with different bifurcation types. The point-point cluster oscillation only relates to pitchfork bifurcation. The point-cycle cluster oscillation includes pitchfork bifurcation and Hopf bifurcation. The point-cycle-cycle cluster oscillation involves three kinds of bifurcation, i.e., the pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation. The larger the amplitude, the more bifurcation types are involved. The research results of cluster oscillation and its generation mechanism will provide valuable theoretical basis for mechanical manufacturing and engineering practice.