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acoustic airfoil algorithm amplitude analysis angle of attack applications approximation assumed asymptotic bifurcation boundary conditions bounds calculations center manifold chaotic coefficients combustion computed considered constant convergence convex set corresponding defined denotes density derivatives determined differential equations diffusion dimensional domain droplet dynamics eigenvalues enclosure methods equa error example exists Figure finite difference fixed flame flow fluid frequency function grid Hopf bifurcation IMACS initial conditions initial value problem input integral interval invariant iteration J.C. Baltzer linear Lyapunov exponent Math mathematical matrix mesh multigrid Navier-Stokes equations nonlinear obtained optimal trajectories ordinary differential equations oscillations paper parameters partial differential equation perturbation phase space physical plane change propagation properties quadratic satisfied Scientific Publishing Section simulation solved sound speed space stability steady stochastic surface Theorem tion transformation variables vector velocity viscous vorticity W.F. Ames editor wave windshear zero