## Bifurcations and their normal form coefficients

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 [25]):

- fold, also known as limit point. We will denote this bifurcation by `LP`
- Hopf-point, denoted by `H`

The equilibrium curve can also have branch points. These are denoted with `BP`. To detect these singularities, we first define 3 test functions:

|  |  | (35)
|  |  | (36)
|  |  | (37)

where is the bialternate matrix product and , are vectors chosen so that the square matrix in (36) is non-singular. (the bialternate matrix product was introduced by C. Stéphanos [36]; see [22] for details). Using these test functions we can define the singularities:

- `BP`:
- `H`:
- `LP`:

A proof that these test functions correctly detect the mentioned singularities can be found in [25]. Here we only notice that not only at Hopf points but also at neutral saddles, i.e. points where 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 . Mathematically, the limit point is nondegenerate (i.e. the equilibrium branch looks locally like a parabola) if and only if In practice, because of round-off errors the computed is always nonzero. So it is more important to check if the testfunction changes sign. (In a cusp point (CP) but this will not be detected on a branch of equilibria since it is a codimension 2 phenomenon, see §8.1)

At a Hopf bifurcation point, the first Lyapunov coefficient is computed by the formula

| (40)

where .

The first Lyapunov coefficient is quite important. If then the Hopf bifurcation is supercritical, i.e. within the center manifold on one side of the bifurcation only stable equilibria exist, on the other side unstable equilibria coexist with stable periodic orbits. If then the Hopf bifurcation is subcritical, i.e. within the center manifold on one side of the bifurcation stable equilibria coexist with unstable periodic orbits, on the other side only unstable equilibria exist. (In a Generalized Hopf point (GH) but this will not be detected on a branch of equilibria since it is a codimension 2 phenomenon, see §8.2)

Subsections

- Branch point locator