Local Bifurcation Control of a Forced Single-Degree-of-Freedom Nonlinear System: Saddle-Node Bifurcation

It is well known that saddle-node bifurcations can occur in the steady-state response of a forced single-degree-of-freedom (SDOF) nonlinear system in the cases of primary and superharmonic resonances. This discontinuous or catastrophic bifurcation can lead to jump and hysteresis phenomena, where at a certain interval of the control parameter, two stable attractors exist with an unstable one in between. A feedback control law is designed to control the saddle-node bifurcations taking place in the resonance response, thus removing or delaying the occurrence of jump and hysteresis phenomena. The structure of candidate feedback control law is determined by analyzing the eigenvalues of the modulation equations. It is shown that three types of feedback – linear, nonlinear, and a combination of linear and nonlinear – are adequate for the bifurcation control. Finally, numerical simulations are performed to verify the effectiveness of the proposed feedback control.