 Solution

Introduce the vector $x=(t,y,y^{\prime },\dots ,y^{(k-1)})\in {\mathbb{R}}^{k+1}$. Its autonomous ordinary differential equation iswhere the coordinate indices start at zero. Appending the clock removes the explicit time dependence.

For a vector field with local existence and uniqueness, its flow ${\varphi }_t(x)$ is the solution at elapsed time $t$ starting at $x$. The uniqueness theorem for ordinary differential equations gives ${\varphi }_s({\varphi }_t(x))={\varphi }_{s+t}(x)$ whenever both sides are defined. The orbit is $O(x)=\{{\varphi }_t(x):t\text{ lies in its maximal interval}\}$. For a forward-complete solution its omega-limit set isequivalently the set of limits along sequences $t_n\to \infty$. A homoclinic orbit is a nonconstant complete orbit approaching the same equilibrium as $t\to -\infty$ and $t\to \infty$. This last definition concerns a general autonomous system; the appended clock itself cannot approach an equilibrium.