In continuous-time systems there are eight generic codim 2 bifurcations that can be detected along
a torus curve:
- 1:1 resonance. We will denote this bifurcation by R1
- 2:1 resonance point, denoted by R2
- 3:1 resonance point, denoted by R3
- 4:1 resonance point, denoted by R4
- Fold-Neimarksacker point, denoted by LPNS
- Chenciner point, denoted by CH.
- Flip-Neimarksacker point, denoted by PDNS
- Double Neimarksacker bifurcation point, denoted by NSNS
To detect these singularities, we first define 6 test functions:
where
is computed by solving
![\begin{displaymath}
\left[\begin{array}{c}D-TA(t)+i\theta I\\
\delta_0-\delta_1
\end{array}\right]_Dv_{1M}=0.
\end{displaymath}](img506.png) |
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The normalization of
is done by requiring
where
is the Gauss-Lagrange quadrature coefficient.
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
.
We compute
by solving
can be computed as
.
The computation of
is done by solving
The expression for the normal form coefficient
becomes
In the 7th test function,
is the monodromy matrix.
In the 8th test function,
, restricted to the subspace without the two eigenvalues with smallest norm.
The singularity matrix is:
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