## 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 [5].

During HHS continuation, all bifurcations detected in HomCont are also detected in our implementation. For this, mostly test functions from [5] 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 [5] 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

| (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).