Bifurcations of limit cycles

On a limit cycle curve the following bifurcations can occur The test function for the Period Doubling bifurcation is defined by the following system
$\displaystyle \left\{ \begin{array}{rl}
\dot v(\tau) - T f_u(u,\alpha)v(\tau) +...
... \\
\int_0^1 \langle \psi(\tau),v(\tau)\rangle d\tau & = 1
\end{array} \right.$     (48)

here $\varphi$ and $\psi$ are so-called bordering vector-functions [26], see [15] for details on the implementation. The system is discretized using orthogonal collocation and solved using the standard MATLAB sparse system solver. The solution component $G \in \ensuremath{\mathbf{R}}$ of this system is the test function and equals zero when there is a Period Doubling bifurcation.

The Fold bifurcation is detected in the same way as the Fold bifurcation of equilibria, the last component of the tangent vector (the $\alpha$ component) is used as the test function.

The Neimark-Sacker bifurcation is detected by monitoring the eigenvalues of the monodromy matrix for the cycle. The monodromy matrix is computed like in AUTO by a block elimination in the discretized form of the Jacobian of (47).

BPC cycles are not generic in families of limit cycles, but they are common in the case of symmetries, if the branch parameter is also the continuation parameter. CL_MATCONT uses a strategy that requires only the solution of linear systems; it is based on the fact that in a symmetry-breaking BPC cycle $M_D$ has rank defect two, where $M_D$ is the square matrix $M_D$, obtained from the discretized form of the Jacobian of (47). To be precise, if $h \in {\cal C}^1([0,1],{\mathrm{I\!R}}^n)$, then

\begin{displaymath}
Mh=
\left[
\begin{array}{c}
\dot{h}-Tf_x(x(t),\alpha)h \\
h(0)-h(1)
\end{array}
\right],
\end{displaymath}

and

\begin{displaymath}
M_D(h)_{dm}=
\left[
\begin{array}{c}
(\dot{h}-Tf_x(x(t),\alpha)h)_{dc} \\
h(0)-h(1)
\end{array}
\right],
\end{displaymath}

where $()_{dm}$ and $()_{dc}$ denote discretization in mesh points and in collocation points, respectively. Therefore we border $M_D$ with two additional rows and columns to obtain

\begin{displaymath}
M_{Dbb}=
\left[
\begin{array}{ccc}
M_D & w_1 & w_2 \\
v_1^* & 0 & 0\\
v_2^* & 0 & 0
\end{array}
\right],
\end{displaymath}

so that $M_{Dbb}$ is nonsingular in the BPC cycle. Then we solve the systems

\begin{displaymath}
M_{Dbb}
\left[
\begin{array}{cc}
\psi_{11} & \psi_{12} \...
...n} & 0_{(Nm+1)n} \\
1 & 0 \\
0 & 1
\end{array}
\right],
\end{displaymath}

where $\psi_{11},\psi_{12}$ have $(Nm+1)n$ components, and $g_{BPC11}$, $g_{BPC12}$, $g_{BPC21}$, and $g_{BPC22}$, are scalar test functions for the BPC. In the BPC cycle they all vanish.

The singularity matrix is

$\displaystyle S = \left(\begin{array}{cccccccc}
0 & 0 & 0 & 0 & - & - & - & - \...
... - & - & - & - & - & 0 & - \\
- & - & - & - & 1 & - & 1 & 0
\end{array}\right)$     (49)

The first row corresponds to the BPC. It contains $4$ zeros which indicates that $g_{BPC11}$, $g_{BPC12}$, $g_{BPC21}$, and $g_{BPC22}$ should vanish. The last row corresponds to the NS. Because we have to exclude that all four testfunctions of the BPC are zeros, we introduce an extra testfunction which corresponds to the norm of these four testfunctions. A NS is detected if this norm is nonzero,the testfunction for the fold is nonzero and the testfunction for the NS is equal to zero.

Subsections