Mathematical definition

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 $G=0$ to (47) where $G$ is the Fold test function. The complete BVP defining a LPC point using the minimal extended system is

$\displaystyle \left\{ \begin{array}{ll}
\frac{du}{dt} - Tf(u,\alpha) & = 0 \\
...
...t),\dot u_{old}(t) \rangle dt & = 0 \\
G[u,T,\alpha] & = 0
\end{array} \right.$     (70)

where $G$ is defined by requiring
\begin{displaymath}
N^1\left(
\begin{array}{c}
v\\
S \\
G
\end{array}...
...\begin{array}{c}
0 \\
0\\
0\\
1
\end{array}\right).
\end{displaymath} (71)

Here $v$ is a function, $S$ and $G$ 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} (72)

where the bordering functions $v_{01},w_{01}$, vector $w_{02}$ and scalars $v_{02}$ and $w_{03}$ are chosen so that $N^1$ 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.