A Fold bifurcation of limit cycles (Limit Point of Cycles, LPC) generically corresponds to a turning point of a curve
of limit cycles. It can be characterized by adding an extra constraint
to (47) where
is the Fold test function. The complete BVP defining a LPC point using the minimal extended system is
![$\displaystyle \left\{ \begin{array}{ll}
\frac{du}{dt} - Tf(u,\alpha) & = 0 \\
...
...t),\dot u_{old}(t) \rangle dt & = 0 \\
G[u,T,\alpha] & = 0
\end{array} \right.$](img438.png) |
|
|
(70) |
where
is defined by requiring
 |
(71) |
Here
is a function,
and
are scalars and
![\begin{displaymath}
N^1 =
\left[
\begin{array}{ccc}
D-Tf_u(u(t),\alpha) &~~...
...{\em Int}_{v_{01}} &~~~ v_{02} &~~~ 0
\end{array}
\right]
\end{displaymath}](img440.png) |
(72) |
where the bordering functions
, vector
and scalars
and
are chosen so that
is nonsingular [15].
This method (using system (71) and (72)) is implemented in the curve definition file limitpointcycle. The discretization is done using orthogonal collocation in the same way as it was done for limit cycles.