Bifurcations along a Neimark-Sacker curve

In continuous-time systems there are eight generic codim 2 bifurcations that can be detected along a torus curve: To detect these singularities, we first define 6 test functions: where $v_{1M}$ 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} (78)

The normalization of $v_{1M}$ is done by requiring $\sum_{i=1}^N\sum_{j=0}^m\sigma_j\langle(v_{1M})_{ij},(v_{1M})_{ij}\rangle t_i=1$ where $\sigma_j$ is the Gauss-Lagrange quadrature coefficient. By discretization we obtain

\begin{displaymath}(v_{1W}^*)^H\left[\begin{array}{c}D-TA(t)+i\theta \\ \delta_0+\delta_1\end{array}\right]_{disc}=0.\end{displaymath}

To normalize $(v_1^*)_{W_1}$ we require $\sum_{i=1}^N\sum_{j=1}^m\left\vert((v_1^*)_{W_1})_{ij}\right\vert _1=1$. Then $\int_{0}^{1} {\langle v_1^*(t),v_1(t)\rangle dt}$ is approximated by $(v_1^*)^T_{W_1}L_{C\times M}v_{1M}$ and if this quantity is nonzero, $v^*_{1W}$ is rescaled so that $\int_{0}^{1} {\langle v_1^*(t),v_1(t)\rangle dt}=1$. We compute $\varphi_{1W}^*$ by solving

\begin{displaymath}(\varphi_{1W}^*)^T\left[\begin{array}{c}D-TA(t)\\ \delta_0-\delta_1\end{array}\right]_{disc}=0\end{displaymath}

and normalize $\varphi_{1W_1}^*$ by requiring $\sum_{i=1}^N\sum_{j=1}^m\left\vert((\varphi_1^*)_{W_1})_{ij}\right\vert _1=1$. Then $\int_{0}^{1} {\langle \varphi_1^*(t),F(u_{0,1}(t))\rangle dt}$ is approximated by $(\varphi_1^*)^T_{W_1}(F(u_{0,1}(t)))_C$ and if this quantity is nonzero, $\varphi^*_{1W}$ is rescaled so that $\int_{0}^{1} {\langle \varphi_1^*(t),F(u_{0,1}(t))\rangle dt}=1$. We compute $h_{20,1M}$ by solving

\begin{displaymath}
\left[\begin{array}{c}D-TA(t)+2i\theta I\\
\delta_0-\delt...
...{array}{c}B(t,v_{1M}(t),v_{1M}(t))\\
0
\end{array}\right].
\end{displaymath}

$a_1$ can be computed as $(\varphi^*_{W_1})^T(B(t,v_{1M},\overline{v}_{1M}))_C$.

The computation of $(h_{11,1})_M$ is done by solving

\begin{displaymath}
\left[\begin{array}{c}
(D-TA(t))_{C\times M}\\
\delta(0)...
...))_C - a_1(F(u_{0,1}(t)))_C\\
0\\
0
\end{array}
\right]
\end{displaymath}

The expression for the normal form coefficient $d$ becomes

\begin{displaymath}
\begin{array}{c}
d= \frac{1}{2}((v_{1W_1}^*)^T,(B(t;h_{11,...
...1}^*)^T(A(t)v_{1}(t))_C+ \frac{ia_1\theta}{T^2}.
\end{array}
\end{displaymath}

In the 7th test function, $M$ is the monodromy matrix.

In the 8th test function, $M2$ is the $(n-2) \times (n-2)$ matrix which restricts the $n \times n$ matrix $M$ to the subspace orthogonal to the two-dimensional left eigenspace of the Neimark-Sacker eigenvalues.

The singularity matrix is:

\begin{displaymath}
S = \left(\begin{array}{cccccc}
0 & - & - & - & - & -\\
...
...- & - & 0 & -\\
- & - & - & - & 1 & 0
\end{array}\right).
\end{displaymath} (79)