The dynamics of a nonlinear structure with a nonlinear magnetic actuator are investigated. A single modal equation of motion is used to analyze the qualitative behaviors of the system. The effects of the feedback gains on the existence of the fractal basin boundaries and the region of the stability basins are considered. For the autonomous case, the resulting third-order system exhibits codimension two bifurcations in which pitchfork bifurcation, saddle connection bifurcation, and Hopf bifurcation exist. When the system is subjected to external disturbance, the Melnikov method was used to show the existence of chaotic motion and the fractal basin boundaries. Also, the global bifurcations in the parametric space are shown. The effects of feedback gains on the system is clearly illustrated by combining these two bifurcation structures. The numerical simulations are performed to verify our analytical results.