Pitch-fork bifurcation

Quadfurcation ) With tongue in cheek, we pointed out that the pitch-fork bifurcation could be called a trifurcation, since three branches of fixed points appear for r > 0. Can you construct an example of a quadfurcation, in which X = /(x, r) has no fixed points for r < 0 and four branches of fixed points for r > 0 Extend your results to the case of an arbitrary number of branches, if possible. 

C.6. 76. Using a computer detect the bifurcation curve in the (a, b)-parameter plane that corresponds to the pitch-fork bifurcation of a symmetric... 

The last comment on the Chua circuit concerns the bifurcations along the path 6 = 6 (see Fig. C.7.4). Notice that this sequence is very typical for many synunetric systems with saddle equilibrium states. We follow the stable periodic orbit starting from the super-critical Andronov-Hopf bifurcation of the non-trivial equilibrium states at a 3.908. As a increases, both separatrices tend to the stable periodic orbits. The last ones go through the pitch-fork bifurcations at a 4.496 and change into saddle type. Their size increases and at a 5.111, they merge with the homoclinic-8. This, as well as subsequent bifurcations, lead to the appearance of the strange attractor known as the double-scroll Chua s attractor in the Chua circuit. ... 

Fig. C.2.2. The Andronov-Hopf bifurcation curve AH and a pitch-fork curve r = 1 in the (r, (r)-plane of the Lorenz model at 6 = 8/3.

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