Hopf bifurcation degenerate

Table 5.1 gives the location of the / 2 = 0 point for various values of y. Points such as this, where the nature of the limit cycle changes qualitatively, are known as degenerate Hopf bifurcations . 

Location of degenerate Hopf bifurcation points from eqn (5.55)... 

Fig. 8.11. The locus H of degenerate Hopf bifurcation points described by the transversality condition (merging of two Hopf points), eqn (8.51). Below this curve, the stationary-state locus exhibits Hopf bifurcation (dynamic instability) at some residence times above it, the system does...

Fig. 8.12. The loci DH, and DH2 corresponding to degenerate Hopf bifurcation points at which the stability of the emerging limit cycle is changing. Again, these are shown relative to the loci for stationary-state multiplicity (broken curves).

Golubitsky, M. and Langford, W. F. (1981). Classification and unfoldings of degenerate Hopf bifurcations. J. Differ. Equations, 41, 373-415. 

This degenerate case typically arises when a nonconservative system suddenly becomes conservative at the bifurcation point. Then the fixed point becomes a nonlinear center, rather than the weak spiral required by a Hopf bifurcation. See Exercise 8.2.11 for another example. 

The degenerate Hopf bifurcation takes place when the term r3 is missing from the standard form of the Hopf bifurcation (5.79). The sensitive state is the same as that for the Hopf bifurcation one pair of purely imaginary eigenvalues ip. 

S. B. Margolis and B. J. Matkowsky, New modes of quasi-periodic combustion near a degenerate Hopf bifurcation point, SIAM J. Appl. Math., 48 (1988), pp. 828-853. 

The stochastic aspect of a complex bifurcation arising in a two variables chemical system is studied. The dynamics reduces, in a suitable region of the phase space, to a normal form for which both roots of the characteristic equation vanish simultaneously. In conditions close to this degenerate situation, the normal form can be viewed as a perturbation of an exactly soluble hamiltonian system, of hamiltonian h, which exhibits a homoclinic trajectory, h = 0. BAESENS and IMICOLIS [l ] have shown that the phase portrait of the dissipative sytem displays two steady states that coalesce, a focus F and a saddle S. [ Moreover, as one moves in the parameter space, a limit cycle surrounding F, bifurcates from a homoclinic trajectory and then disappears by Hopf bifurcation. ... 

Two other points are marked, one along each Hopf curve. These are the degenerate bifurcation points at which the emerging limit cycle changes from stable (supercritical) to unstable (subcritical). These have the locations... 

---