Cluster oscillation and bifurcation of fractional-order Duffing system with two time scales

The dynamical behavior on fractional-order Duffing system with two time scales is investigated, and the point-cycle coupling type cluster oscillation is firstly observed herein. When taking the fractional order as bifurcation parameter, the dynamics of the autonomous Duffing system will become more complex than the corresponding integer-order one, and some typical phenomenon exist only in the fractional-order one. Different attractors exist in various parameter space, and Hopf bifurcation only happens while fractional order is bigger than 1 under certain parameter condition. Moreover, the bifurcation behavior of the autonomous system may regulate dynamical phenomenon of the periodic excited system. It results into the point-cycle coupling type cluster oscillation when the fractional order is bigger than 1. The related generation mechanism based on slow-fast analysis method is that the slow variation of periodic excitation makes the system periodically visit different attractors and critical points of different bifurcations of the autonomous system. Graphic abstract