To protect the aircraft flight safety across the envelope of angle of attack, bifurcation analysis and backstepping method are investigated to predict and suppress the unstable nonlinear flight phenomena. By applying bifurcation analysis and continuation method to the flight motion, the onsets of both the chaotic phenomenon called falling leaf motion and the catastrophe phenomenon named coupled jump behavior are derived. To stabilize these unstable motions, a backstepping and disturbance observer based flight controller is designed. According to the main function of the control surface, we divide the flight controller into the airspeed subsystem and the flight path subsystem, where the airspeed subsystem is regulated by an adaptive dynamic inversion controller while the flight path subsystem is stabilized by a third-order compound controller. Considering the parametric uncertainties of aerodynamics, three sliding mode disturbance observers are presented as compensators to approximate the compound uncertainties. Simulations demonstrate that the proposed controller can recover the aircraft from falling leaf motion or coupled jump behavior to straight level flying.