In CL_MATCONT there are five generic codim 2 bifurcations that can be detected along
a fold of cycles curve:
- Branch point. We will denote this bifurcation with BPC
- Strong 1:1 resonance. We will denote this bifurcation with R1
- Cusp point, denoted by CPC
- Fold - flip, Limit Point - Period Doublingor We will denote this bifurcation by LPPD
- Fold - Neimark-Sacker, denoted by LPNS
A Generalized Hopf (GH) marks the end (or start) of an LPC curve.
To detect the generic singularities, we first define
test functions, where
is the number of branch parameters:
-
-
-
(normal form coefficient)
-
-
In the
expressions
is the vector computed in (57) and
(branch parameter) is a component of
.
In the second expression
, we compute
by solving
![\begin{displaymath}
\left[\begin{array}{c}D-TA(t)\\
\delta_0-\delta_1\\
\in...
...y}{c}TF(u_{0,1}(t))\\
0\\
0
\end{array}\right]_{disc}.
\end{displaymath}](img454.png) |
(73) |
By discretization we obtain
To normalize
we require
. Then
is approximated by
and if this quantity is nonzero,
is rescaled so that
.
So the third expression for the normal form coefficient
becomes
In the fourth expression,
is the monodromy matrix.
In the fifth expression,
, restricted to the subspace without the two eigenvalues with smallest norm.
The number of branch parameters is not fixed. If the number of branch parameters is
then this matrix has three more rows and columns. This singularity matrix is automatically extended: