In continuous-time systems there are two generic codim 1 bifurcations that can be detected along the equilibrium curve (no derivations will be done here; for more detailed information see [10]):
The equilibrium curve can also have branching points. These are denoted with BP.
To detect these singularities, we first define 3 test functions:
|
where \odot is the bialternate matrix product. Using these test functions we can define the singularities:
A proof that these test functions correctly detect the mentioned singularities can be
found in [10]. Here we only notice that f2=0 not only at Hopf points
but also at neutral saddles, i.e. points where fx has two real eigenvalues with sum zero. So, the singularity matrix is:
| (38) |
For each detected limit point, the corresponding quadratic normal form
coefficient is computed:
| (39) |
where fu q=fTu p = 0, pTq = 1. At a Hopf bifurcation
point, the first Lyapunov coefficient is computed by the formula
| (40) |
where fuq=iwq, fTu p=-iwp, [`p]Tq=1.