This paper presents a methodology for systematically studying the nonlinear frequency responses of an aircraft model using numerical continuation with periodic forcing, thereby presenting an extension of conventional bifurcation analysis in flight dynamics applications. The motivation is to identify nonlinear phenomena in the frequency domain that are absent in linearized models - upon which many control law designs are based - and which therefore risks degrading the performance or robustness of the linear-model based controllers. Since the aerospace industry typically uses linearizations in controller design, both open and closed loop behaviors are considered. When the example aircraft considered here is forced with small control surface deflections, highly nonlinear responses are observed. This includes period doubling bifurcations, fold bifurcations leading to existence of multiple solutions, quasi periodic motions, and formation of isolas. Closed-loop responses of a proportional stability augmentation controller for this aircraft become out of phase with the linear prediction at low forcing frequencies when the aircraft operates at high angle of attack. To address these behaviors, the methodology is extended by employing two-parameter continuation of the controller gain to assess its effectiveness in those nonlinear regions, where linear controller design techniques cannot be used. Time histories are used to verify the results.