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This paper investigates stability analysis of flapping flight. Due to time-varying aerodynamic forces, such systems do not display fixed points of equilibrium. The problem is therefore approached via a limit cycle analysis based on Floquet theory. Stability is assessed from the eigenvalues of the Jacobian matrix associated to the limit cycle, also known as the Floquet multipliers. We developed this framework to analyze the flapping flight equations of motion of a bird in the longitudinal plane. Such a system is known to be not only non-linear and time-dependent, but also driven by state-dependent forcing aerodynamic forces. A model accounting for wing morphing under prescribed kinematics is developed for generating realistic state-dependent aerodynamic forces. The morphing wing geometry results from the envelope of continuously articulated rigid bodies, modeling bones and feather rachises, and capturing biologically relevant degrees of freedom. A sensitivity analysis is carried out which allows studying several flight configurations in trimmed state. Our numerical results show that in such a system one instability mode is ubiquitous, thus suggesting the importance of sensory feedback to achieve steady-state flapping flight in birds. The effect of wingbeat amplitude, governed by the shoulder joint, is found to be crucial in tuning the gait towards level flight, but marginally affects stability. In contrast, the relative position between the wing and the center of mass is found to significantly affect the values of Floquet multipliers, suggesting that the distribution of pitching moment plays a very important role in flapping flight stability.