Cusp and Generalized Hopf bifurcations

Here the bifurcations concern singularities with co-dimension greater than 1. These are only unfolded by local families that depend on more than 1 parameter.

Cusp bifurcation

Given a smooth vector field with a cusp singularity at the origin. Then but == and has normal form (with and ).The local familyis a versal unfolding of the cusp singularity above. (Note that the cusp singularity requires at least two parameters to completely unfold it). Considering this local family , under some conditions (not shown here) in particular about parameters, it is possible to generate three bifurcation types: (1) Saddle node, (2) Hopf bifurcation, (3) Saddle connection.

Generalized Hopf bifurcations

Generalized Hopf bifurcations. (Takens 1973b, Guckenheimer & Holmes 1983, Arrowsmith & Place 1990). Here, we describe the generalized Hopf singularity of type . The vector field in the previous co-dimension 1 Hopf bifurcation was obtained by posing in the normal formIf a degeneracy is introduced by posing ==== with , then the system above is said to have a generalized Hopf singularity of type at =. (Note that a co-dimension Hopf singularity is said to be of type ). The framework of the generalized Hopf singularity highlights the relationship between the degeneracy of a singularity and the number of parameters contained in its versal unfolding.

If a Hopf singularity of type occurs, it is possible to reduce the system above to a simpler system. Giventhat can also be written:with =. Because >, we have > for all enough close to the origin. Thus there exists a neighborhood of = on which the system is topologically equivalent to the (simpler) vector field:While preserving the topological equivalence, we can replace by in the vector field above. Two cases are then possible according to a singularity of the type: or . A versal unfolding for the vector field above can be obtained through the following Takens theorem (1973b):

Theorem

The versal unfolding of a generalized Hopf singularity of type given in the simpler vector field above is written (where , =):

According to the sign of , the unfoldings of in this theorem gives two cases and . Here, we only describe the first case , and the latter case can be related to the other by a time reversal (so the stabilities of fixed points and limit cycles are inverted). The formula in this theorem, through the polar form ( with here =), gives for the bifurcational behavior by using the equivalent equation: The behavior is independent of the terms of order (as described for the co-dimension 1 Hopf bifurcation), so it suffices to consider . This equation in particular provides limit cycles. (1) For =, by posing =, we obtain , then a limit cycle occurs whose radius is only when <. (2) For =, by posing =, we obtain and non-trivial zeros result from . Then we obtain: If > and > (or <): no limit cycle but a repelling spiral. If < and >: two limit cycles. If <: one limit cycle. (see Fig.below).This figure shows the bifurcation diagram for the type-(2,+) Hopf bifurcation. It shows that a supercritical (subcritical) Hopf bifurcation occurs when increases through zero with < (subcritical with >), and a double limit cycle bifurcation occurs on the semi-parabola. If decreases through the semi-parabola, a non-hyperbolic limit cycle occurs and then divides into two hyperbolic cycles.