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}](img508.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,
is the
matrix which restricts the
matrix
to the subspace orthogonal to the two-dimensional left eigenspace of the Neimark-Sacker eigenvalues.
The singularity matrix is:
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