We study the nonlinear behavior of a wind turbine airfoil oscillating along the pitch mode of oscillation. In the analysis, the nonlinearity of the structural material, the airspeed, and the reduced frequency of oscillation are the important parameters considered. The equation of motion for the pitching oscillation case is modeled and derived using the Office National d’Etudes et de Recherches Aérospatiales (ONERA) dynamic stall model. In the ONERA dynamic stall model, the aerodynamic loads are given in terms of differential equations. We focus on the influence of the material structural nonlinearity and nondimensional airspeed on the wind turbine airfoil during the pitch mode of oscillation. The type of airfoil considered is the National Advisory Committee for Aeronautics (NACA) 0012 airfoil. To understand the effects of dynamic stall on the system under different situations of structural nonlinearity, we use bifurcation plots, phase plots, Poincaré maps, and the maximum Lyapunov exponent (MLE) to analyze the behavior of the airfoil and avoid the effect of non-periodic motion of the airfoil on the wind turbine system. The results show that chaos occurs for specific parameters of the airfoil of the wind turbine, which can be avoided by avoiding certain operating conditions corresponding to chaotic motion. This finding can be used as a guideline for wind turbine design and decrease the cost of maintenance.