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
to (47) where
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
![$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) & = 0 \\
...
... x_{old}(t) \rangle dt & = 0 \\
G[x,T,\alpha,\kappa] & = 0
\end{array} \right.$](img484.png) |
|
|
(75) |
where
is defined by requiring
 |
(76) |
Here
and
are functions and
and
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}](img491.png) |
(77) |
where the bordering functions
, vectors
and
are chosen so that
is nonsingular [15]. We note that an additional variable
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
. The additional variable
is introduced as the last continuation variable of neimarksacker.m, after the state variables, the period and the two active system parameters.