Bifurcation Transition and Nonlinear Response in a Fractional Order System

Abstract

We extend a typical system that possesses a transcritical bifurcation to a fractionalorder 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 lowfrequency excitation, the system parameter induces a continuous transcritical bifurcation. When both lowand highfrequency forces are acting, the highfrequency force has a biasing effect and it makes the continuous transcritical bifurcation transit to a discontinuous saddlenode bifurcation. For this case, the system parameter, the highfrequency force, and the fractionalorder have effects on the saddlenode bifurcation. The system parameter induces twice a saddlenode bifurcation. The amplitude of the highfrequency force and the fractionalorder induce only once a saddlenode bifurcation in the subcritical and the supercritical case, respectively. The system presents a nonlinear response to the lowfrequency force. The system parameter and the lowfrequency can induce a resonancelike behavior, though the highfrequency force and the fractionalorder 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 fractionalorder system.