= Solution
Introduce the vector $x=(t,y,y',\ldots,y^{(k-1)})\in\mathbb R^{k+1}$. Its autonomous <ordinary differential equation> is
$$
\dot x=(1,x_2,x_3,\ldots,x_k,g(x_0,x_1,\ldots,x_k)),
$$
where 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> $\phi_t(x)$ is the solution at elapsed time $t$ starting at $x$. The <uniqueness theorem for ordinary differential equations> gives $\phi_s(\phi_t(x))=\phi_{s+t}(x)$ whenever both sides are defined. The <orbit> is $O(x)=\{\phi_t(x):t\text{ lies in its maximal interval}\}$. For a forward-complete solution its <omega-limit set> is
$$
\omega(x)=\bigcap_{T\ge0}\overline{\{\phi_t(x):t\ge T\}},
$$
equivalently 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.
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