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:
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:
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:
![\begin{displaymath}
a=\frac{1}{2} p^Tf_{uu}[q,q],
\end{displaymath}](img206.png) |
(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
![\begin{displaymath}
l_1=\frac{1}{2}{\rm Re}\left\{p^T
\left(f_{uuu}[q,q,\bar{q...
...bar{q},(2i\omega I_n-f_u)^{-1}f_{uu}[q,q]] \right)\right\},
\end{displaymath}](img212.png) |
(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