Consider a double aerodynamic pendulum that represents a double pendulum, the second link of which carries a thin wing with symmetrical airfoil. The entire system is placed in flow with the constant speed V and mounted in such a way that both axes of the pendulum are vertical. We assume that both joints of the pendulum are equipped with linear spiral springs. Systems of rigid bodies that move under the combined action of elastic forces and aerodynamic (or hydrodynamic) load are called aeroelastic systems and are of great interest from the point of view of basic research and applications (especially in the areas of aerospace and civil engineering). An elastically mounted double-link aerodynamic pendulum is considered. It is assumed that the flow acts only upon the second link of the system. Conditions of asymptotic stability of the trivial equilibrium "along the flow" (when both links are stretched along the flow) are obtained. Influence of position of the wing and of coefficients of structural stiffness upon the stability is discussed. Limit cycles are studied that arise in the system for a certain range of values of parameters. Dependence of their amplitude on coefficients of stiffness is analyzed. Experiments with such pendulum are performed in the wind tunnel of the Institute of Mechanics of Lomonosov MSU, where parameters of periodic motions are registered for different wind speeds, and different locations of the wing with respect to the second link. It is shown that experimental data is in qualitative agreement with results of numerical simulation.