Homoclinic initialization (HHS)

For homoclinics the same problems occur as with limit cycles, a homoclinic continuation cannot be done by just calling the continuer as
[x,v,s,h,f]=cont(@homoclinic, x0, v0, opt)
The homoclinic curve file has to know Also, some initial information about the state variables is necessary. In the simplest cases this is an initial cycle x0 with a large period and close to a homoclic orbit, or an already known homoclinic orbit. This information can be supplied using an initializer:

These initializers return an initial homoclinic orbit x0 as well as its tangent vector v0.

It is also possible to start a homoclinic orbit from a Bogdanov-Takens equilibrium point. The initialization of homoclinic orbits from a BT point has a long history. We refer in particular to [30], [31] and [1] where further references can be found. This case is relegated to the MATCONT GUI tutorials.

The homotopy method even allows a start from any equilibrium point. For details and examples we refer to [8]. This case is also relegated to the MATCONT GUI tutorials.