Bifurcations

During HSN continuation, only one bifurcation is tested for, namely the non-central homoclinic-to-saddle-node orbit or NCH. This orbit forms the transition between HHS and HSN curves. The strategy used for detection is taken from HomCont [6].

During HHS continuation, all bifurcations detected in HomCont are also detected in our implementation. For this, mostly test functions from [6] are used.

Suppose that the eigenvalues of $f_x(x_0,\alpha_0)$ can be ordered according to

\begin{displaymath}
\Re(\mu_{ns}) \leq ... \leq \Re(\mu_1) < 0 < \Re(\lambda_1) \leq ... \leq \Re(\lambda_{nu}),
\end{displaymath} (95)

where $\Re()$ stands for 'real part of', $ns$ is the number of stable, and $\nu$ the number of unstable eigenvalues. The test functions for the bifurcations are

For orbit- and inclination-flip bifurcations, we assume the same ordering of the eigenvalues of $f_x(x_0,\alpha_0) = A(x_0,\alpha_0)$ as shown in (95), but also that the leading eigenvalues $\mu_1$ and $\lambda_1$ are unique and real:

\begin{displaymath}
\Re(\mu_{ns}) \leq ... \leq \Re(\mu_2) < \mu_1 < 0 < \lambda_1 < \Re(\lambda_2) \leq ... \leq \Re(\lambda_{nu})\ .
\end{displaymath}

Then it is possible to choose normalised eigenvectors $p_1^s$ and $p_1^u$ of $A^T(x_0,\alpha_0)$ and $q_1^s$ and $q_1^u$ of $A(x_0,\alpha_0)$ depending smoothly on $(x_0,\alpha_0)$, which satisfy
$\displaystyle A^T(x_0,\alpha_0)\ p_1^s = \mu_1\ p_1^s$ $\textstyle \ \ \ \ $ $\displaystyle A^T(x_0,\alpha_0)\ p_1^u = \lambda_1\ p_1^u$  
$\displaystyle A(x_0,\alpha_0)\ q_1^s = \mu_1\ q_1^s$ $\textstyle \ \ \ \ $ $\displaystyle A(x_0,\alpha_0)\ q_1^u = \lambda_1\ q_1^u \ .$  

The test functions for the orbit-flip bifurcations are then:

For the inclination-flip bifurcations, in [6] the following test functions are introduced:

where $\phi$ ( $\phi \in {\cal C}^1([0,1],{\mathbb{R}}^n)$) is the solution to the adjoint system, which can be written as
\begin{displaymath}
\left\{\begin{array}{l} \dot{\phi}(t) + 2\:T\:A^T(x(t),\al...
...i}^T(t)[\phi(t)-\widetilde{\phi}(t)]dt = 0\end{array} \right.
\end{displaymath} (96)

where $L_s$ and $L_u$ are matrices whose columns form bases for the stable and unstable eigenspaces of $A(x_0,\alpha_0)$, respectively, and the last condition selects one solution out of the family $c\phi(t)$ for $c \in {\mathbb{R}}$. $L_u$ is equivalent to $Q_U$ from the mathematical definition of the system, and $L_s$ to $Q_S$. In the homoclinic defining system the orthogonal complements of $Q_S$ and $Q_U$ are used; in the adjoint system for the inclination-flip bifurcation, we use the matrices themselves (or at least, their transposed versions).