In MATCONT / CL_MATCONT there are four generic codim 2 bifurcations that can be detected along
a flip curve:
- Strong 1:2 resonance. We will denote this bifurcation with R2
- Fold - flip, Limit Point - Period Doublingor We will denote this bifurcation with LPPD
- Flip - Neimark-Sacker, denoted as PDNS
- Generalized period doubling point, denoted as GPD
To detect these singularities, we define 4 test functions:
where
is the coefficient defined in (52),
is the monodromy matrix and
is the bialternate product.
The
's and
's are obtained as follows. For a given
we consider three different discretizations:
-
the vector of values in the mesh points,
-
the vector of values in the collocation points,
-
where
is the vector of values in the collocation points multiplied with the Gauss - Legendre weights and the lengths of the mesh intervals, and
.
Formally we further introduce
which is a structured sparse matrix that converts a vector
of values in the mesh points into a vector
of values in the collocation points by
.
We compute
by solving
![\begin{displaymath}
\left[\begin{array}{c}D-TA(t)\\
\delta_0+\delta_1
\end{array}\right]_{disc}v_{1M}=0.
\end{displaymath}](img407.png) |
(67) |
The normalization of
is done by requiring
where
is the Gauss-Lagrange quadrature coefficient and
is the length of the i-th interval.
By discretization we obtain
To normalize
we require
. Then
is approximated by
and if this quantity is nonzero,
is rescaled so that
. We compute
by solving
and normalize
by requiring
. Then
is approximated by
and if this quantity is nonzero,
is rescaled so that
.
can be computed as
.
The computation of
is done by solving
The expression for the normal form coefficient
becomes
The singularity matrix is:
 |
(68) |