Homoclinic orbits: Mathematical definition

In dynamical systems theory, an orbit corresponding to a solution $u(t)$ is called homoclinic to the equilibrium point $u^0$ of the dynamical system if $u(t)
\rightarrow u^0$ as $t \rightarrow \pm\infty$. There are two types of homoclinic orbits with codimension 1, namely homoclinic-to-hyperbolic-saddle (HHS), if $u^0$ is a saddle (saddle-focus or bi-focus), and homoclinic-to-saddle-node (HSN), if $u^0$ is a saddle-node (i.e., exhibits a limit point bifurcation). We recall that AUTO has a toolbox for homoclinic continuation, named HOMCONT [5], [6].

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 [8] into the defining system. For some details on the implementation of the homoclinic continuation we refer to [19].



Subsections