Saddle-Node Bifurcation and Vibrational Resonance in a Fractional System with an Asymmetric Bistable Potential

We investigate the saddle-node bifurcation and vibrational resonance in a fractional system that has an asymmetric bistable potential. Due to the asymmetric nature of the potential function, the response and its amplitude closely depend on the potential well where the motion takes place. And consequently for numerical simulations, the initial condition is a key and important factor. To overcome this technical problem, a method is proposed to calculate the bifurcation and response amplitude numerically. The numerical results are in good agreement with the analytical predictions, indicating the validity of the numerical and theoretical analysis. The results show that the fractional-order of the fractional system induces one saddle-node bifurcation, while the asymmetric parameter associated to the asymmetric nature of the potential function induces two saddle-node bifurcations. When the asymmetric parameter vanishes, the saddle-node bifurcation turns into a pitchfork bifurcation. There are three kinds of vibrational resonance existing in the system. The first one is induced by the high-frequency signal. The second one is induced by the fractional-order. The third one is induced by the asymmetric parameter. We believe that the method and the results shown in this paper might be helpful for the analysis of the response problem of nonlinear dynamical systems.