Mathematical definition

A torus bifurcation of limit cycles (Neimark-Sacker, NS) generically corresponds to a bifurcation to an invariant torus, on which the flow contains periodic or quasi-periodic motion. It can be characterized by adding an extra constraint $G=0$ to (47) where $G$ is the torus test function which has four components from which two are selected. The complete BVP defining a NS point using a minimally extended system is

$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) & = 0 \\
...
... x_{old}(t) \rangle dt & = 0 \\
G[x,T,\alpha,\kappa] & = 0
\end{array} \right.$     (75)

where

\begin{displaymath}
G=\left(\begin{array}{cc}
G^{11}&G^{12}\\
G^{21}&G^{22}
\end{array}\right)\end{displaymath}

is defined by requiring
\begin{displaymath}
N^3\left(
\begin{array}{cc}
v^1&v^2\\
G^{11}&G^{12} \...
...
0&~~0 \\
0&~~0\\
1&~~0\\
0&~~1
\end{array}\right).
\end{displaymath} (76)

Here $v^1$ and $v^2$ are functions and $G^{11},G^{12},G^{21}$ and $G^{22}$ are scalars and
\begin{displaymath}
N^3 =
\left[
\begin{array}{ccc}
D-Tf_x(x(\cdot),\alpha)...
...\mbox{\em Int}_{v_{02}} &~~~ 0 &~~~ 0
\end{array}
\right]
\end{displaymath} (77)

where the bordering functions $v_{01},v_{02},w_{11},w_{12}$, vectors $w_{21}$and $w_{22}$ are chosen so that $N^3$ is nonsingular [15]. We note that an additional variable $\kappa$ is introduced in (75). This method (using system (76) and (77)) is implemented in the curve definition file neimarksacker.m. The discretization is done using orthogonal collocation over the interval $[0\ 2]$. The additional variable $\kappa$ is introduced as the last continuation variable of neimarksacker.m, after the state variables, the period and the two active system parameters.