Bifurcation Transition and Nonlinear Response in a Fractional-Order System

Abstract

We extend a typical system that possesses a transcritical bifurcation to a fractional-order version. The bifurcation and the resonance phenomenon in the considered system are investigated by both analytical and numerical methods. In the absence of external excitations or simply considering only one low-frequency excitation, the system parameter induces a continuous transcritical bifurcation. When both low- and high-frequency forces are acting, the high-frequency force has a biasing effect and it makes the continuous transcritical bifurcation transit to a discontinuous saddle-node bifurcation. For this case, the system parameter, the high-frequency force, and the fractional-order have effects on the saddle-node bifurcation. The system parameter induces twice a saddle-node bifurcation. The amplitude of the high-frequency force and the fractional-order induce only once a saddle-node bifurcation in the subcritical and the supercritical case, respectively. The system presents a nonlinear response to the low-frequency force. The system parameter and the low-frequency can induce a resonance-like behavior, though the high-frequency force and the fractional-order cannot induce it. We believe that the results of this paper might contribute to a better understanding of the bifurcation and resonance in the excited fractional-order system.