## An analytical and numerical study of axial flow compressor instability |

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aperiodic behaviour axial mass flow axisymmetric characteristic axisymmetric fixed point axisymmetric limit cycle axisymmetric solution bifurcation diagram bifurcation occurs bifurcation set bifurcation theory calculations centre manifold Compression Systems compressor face cubic function degenerate Hopf bifurcation depicted in figure dimensional discussed double cycles dynamics eigenvalue equilibrium mass flow equilibrium point Fourier mode guide vane loss homoclinic orbits Hopf bifurcation points initial condition inlet duct inlet guide vane Jacobian derivative Jordan normal form line of Hopf loses stability mass flow coefficient negative normal form observed parameter space parameter values partial differential equations percentage of energy periodic orbit periodic solution perturbations plane qualitative rescaled Rotating stall cell rotating stall characteristic rotating stall disturbance rotating stall point rotating stall solution saddle loop saddle-node bifurcation set of equations shown in figure steady axisymmetric flow subcritical supercritical Takens-Bogdanov bifurcation Takens-Bogdanov point throttle is closed throttle settings transcritical bifurcation values of equilibrium wheel speed zero