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 $\QTR{bf}{X}_{0}$ a smooth vector field with a cusp singularity at the origin. Then MATH but MATH=MATH=$0$ and MATH has normal form (with $a_{2}\neq 0$ and $b_{2}\neq 0$).MATHThe local familyMATHis 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 MATH, 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 $p$. The vector field in the previous co-dimension 1 Hopf bifurcation was obtained by posing $a_{1}\neq 0$ in the normal formMATHIf a degeneracy is introduced by posing $a_{1}$=$a_{2}$=$\cdots $=$a_{p-1}$=$0$ with $a_{p}\neq 0$, then the system above is said to have a generalized Hopf singularity of type $p$ at $(x_{1},x_{2})^{T}$=$\QTR{bf}{0}$. (Note that a co-dimension $1$ Hopf singularity is said to be of type $1$). 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 $p$ occurs, it is possible to reduce the system above to a simpler system. GivenMATHthat can also be written:MATHwith $\gamma _{1}$=$a_{1}/\alpha $. Because $\alpha $>$0$, we have $\phi (x_{1},x_{2})$>$0$ for all $(x_{1},x_{2})$ enough close to the origin. Thus there exists a neighborhood of $(x_{1},x_{2})^{T}$=$\QTR{bf}{0}$ on which the system is topologically equivalent to the (simpler) vector field:MATHWhile preserving the topological equivalence, we can replace $\gamma _{p}$ by MATH in the vector field above. Two cases are then possible according to a singularity of the type: $(p,+)$ or $(p,-)$. A versal unfolding for the vector field above can be obtained through the following Takens theorem (1973b):


The versal unfolding of a generalized Hopf singularity of type $p$ given in the simpler vector field above is written (where $s_{i}\in R$, $i$=$0,1,..,p-1$):MATH

According to the sign of $\gamma _{p}$, the unfoldings of in this theorem gives two cases $(p,+)$ and $(p,-)$. Here, we only describe the first case $(p,+)$, 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 (MATH with here $\dot{\theta}$=$1$), gives for $(p,+)$ the bifurcational behavior by using the equivalent equation: MATH The behavior is independent of the terms of order $r^{2p+3}$ (as described for the co-dimension 1 Hopf bifurcation), so it suffices to consider MATH. This equation in particular provides limit cycles. (1) For $p$=$1$, by posing $r^{2}$=$\lambda $, we obtain MATH, then a limit cycle occurs whose radius is $(-s_{0})^{1/2}$ only when $s_{0}$<$0$. (2) For $p$=$2$, by posing $r^{2}$=$\lambda $, we obtain MATH and non-trivial zeros result from MATH. Then we obtain: $(i)$ If $s_{0}$>$0$ and $s_{1}$>$0$ (or $s_{1}^{2}-4s_{0}$<$0$): no limit cycle but a repelling spiral. $(ii)$ If $s_{1}$<$0$ and $s_{1}^{2}-4s_{0}$>$0$: two limit cycles. $(iii)$ If $s_{0}$<$0$: one limit cycle. (see Fig.$\,$below).MATHThis figure shows the bifurcation diagram for the type-(2,+) Hopf bifurcation. It shows that a supercritical (subcritical) Hopf bifurcation occurs when $s_{0}$ increases through zero with $s_{1}$<$0$ (subcritical with $s_{1}$>$0$), and a double limit cycle bifurcation occurs on the semi-parabola. If $s_{0}$ decreases through the semi-parabola, a non-hyperbolic limit cycle occurs and then divides into two hyperbolic cycles.