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
can be ordered according to
 |
(95) |
where
stands for 'real part of',
is the number of stable, and
the number of unstable eigenvalues.
The test functions for the bifurcations are
- Neutral saddle, saddle-focus or bi-focus
If both
and
are real, then it is a neutral saddle, if one is real and one consists of a pair of complex conjugates, the bifurcation is a saddle-focus, and it is a bi-focus when both eigenvalues consist of a pair of complex conjugates.
- Double real stable leading eigenvalue
- Double real unstable leading eigenvalue
- Neutrally-divergent saddle-focus (stable)
- Neutrally-divergent saddle-focus (unstable)
- Three leading eigenvalues (stable)
- Three leading eigenvalues (unstable)
- Non-central homoclinic-to-saddle-node
- Shil'nikov-Hopf
- Bogdanov-Takens point
For orbit- and inclination-flip bifurcations, we assume the same ordering of the eigenvalues of
as shown in (95), but also that the leading eigenvalues
and
are unique and real:
Then it is possible to choose normalised eigenvectors
and
of
and
and
of
depending smoothly on
, which satisfy
The test functions for the orbit-flip bifurcations are then:
- Orbit-flip with respect to the stable manifold
- Orbit-flip with respect to the unstable manifold
For the inclination-flip bifurcations, in [6] the following test functions are introduced:
- Inclination-flip with respect to the stable manifold
- Inclination-flip with respect to the unstable manifold
where
(
) 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}](img682.png) |
(96) |
where
and
are matrices whose columns form bases for the stable and unstable eigenspaces of
, respectively, and the last condition selects one solution out of the family
for
.
is equivalent to
from the mathematical definition of the system, and
to
. In the homoclinic defining system the orthogonal complements of
and
are used; in the adjoint system for the inclination-flip bifurcation, we use the matrices themselves (or at least, their transposed versions).