Dynamic soaring phenomenon, exhibited by soaring birds, has long been a biological inspiration for the aerospace and control engineering communities. If this fascinating phenomenon, which allows soaring birds to perform almost un-powered flights using wind shear, can be mimicked by Unmanned-Aerial-Vehicles (UAVs), then UAVs performance have a substantial potential to enhance technologically and economically as well. Even though there have been considerable amount of research covering the modeling, optimization, control and simulation aspects of different UAVs performing dynamic soaring, there is little to non conclusive work analyzing the stability of such UAVs about the soaring orbits. In this paper, we present a comprehensive framework for determining the stability of soaring UAVs utilizing both linear (Floquet-based) and nonlinear (Contraction-theory-based) techniques. Stability analysis under Floquet remained inconclusive , which provoked nonlinear Contraction formulation in order to reach a conclusive stability assessment for the actual nonlinear fixed-wing UAV performing dynamic soaring. Furthermore, parametric variation along with numerical simulations were conducted to ascertain the response of the actual nonlinear system when perturbed from its nominal motion studied in this paper. The analysis and simulations revealed that the system posses instability as the UAV motion diverges from its nominal trajectory and follow an undesirable path. By this result, we conclude, for the first time in literature, that UAVs performing dynamic soaring in an optimal way to reduce wind shear requirements, are inherently unstable. The results of this paper suggests that mimicking dynamic soaring by UAVs, require careful investigations of tracking and regulatory controls that should be implemented.