Global Analysis of Limit Cycle Oscillations

The Hopf bifurcation theorem is only valid in the neighborhood of the bifurcation value n = 4. However, it is possible to prove existence of a limit cycle for finite values n 4, and uniqueness and stability for infinite n. 

I have only been able to establish uniqueness and stability of the periodic solution for Eq. (27) for the case where n is infinite. For this case, the 

Consider a trajectory through any point (Xq, Yo o) th volume 000. From the previous section, the equation of the trajectory through this point in volume 000 is 

Since the trajectories in 000 and 100 can be superimposed under a symmetry operation of the cube, it can be seen that there will be a fixed point provided 

Substituting these values in Eq. (31) and eliminating Yq from the resulting equation gives a cubic equation in XqI 

---