Chaotic motions of a two-dimensional airfoil with cubic nonlinearity in supersonic flow

Using a combination of analytical and numerical methods, the paper studies chaotic motions of a two-dimensional airfoil with cubic nonlinearity in supersonic flow. By using the undetermined coefficient method, the homoclinic orbit is found, and the uniform convergence of the homoclinic orbit series expansion is proved. It analytically demonstrates that there exists a homoclinic orbit joining the initial equilibrium point to itself, therefore Smale horseshoe chaos occurs for this system via Siʼlnikov criterion. The system evolves into chaotic motion through period-doubling bifurcation, and is periodic again as the dimensionless airflow speed increases. While the system is periodic and chaotic alternately as the wing mass static moment about the elastic axis increases. When the nonlinear stiffness coefficient crosses its critical value, the system is always chaotic. Numerical simulations are also given, which confirm the analytical results.