## Homoclinic orbits: Mathematical definition

In dynamical systems theory, an orbit corresponding to a solution is called homoclinic to the equilibrium point of the dynamical system if as . There are two types of homoclinic orbits with codimension 1, namely homoclinic-to-hyperbolic-saddle (HHS), whereby is a saddle, saddle-focus or bi-focus, and homoclinic-to-saddle-node (HSN), whereby is a saddle-node (i.e., exhibits a limit point bifurcation). We recall that AUTO has a toolbox for homoclinic continuation, named HOMCONT [4], [5].

During the continuation, it is necessary to keep track of several eigenspaces of the equilibrium in each step. To do this in an efficient way, MATCONT incorporates the continuation of invariant subspaces [7] into the defining system.

Subsections

- Homoclinic-to-Hyperbolic-Saddle Orbits
- Homoclinic-to-Saddle-Node Orbits