Tangent double Hopf bifurcation in a differentially rotating cylinder flow

A tangent double Hopf bifurcation has been found in a rotating cylinder flow driven by the counter-rotation of the top endwall. The codimension-3 tangent double Hopf point has been located using linear stability analysis. The nonlinear analysis of the multiple solutions in a neighborhood of this bifurcation point is performed by numerical solutions of the three-dimensional Navier-Stokes equations. At the tangent double Hopf point, two rotating waves and an unstable modulated rotating wave bifurcate simultaneously. A center manifold/normal form analysis is also performed, which is in very good agreement with both the linear and nonlinear computations. By a detailed exploration of the nonlinear flow, we have unraveled the complex dynamics generated by the mode competition, which is organized by the codimension-3 tangent double Hopf bifurcation.