Given a non-autonomous $k$ th-order differential equation

$$\frac{d^{k} y}{d t^{k}}=g\left(t, y, \frac{d y}{d t}, \frac{d^{2} y}{d t^{2}}, \ldots, \frac{d^{k-1} y}{d t^{k-1}}\right)$$

with $y \in \mathbb{R}$, explain how it may be written in the autonomous first-order form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ for suitably chosen vectors $\mathbf{x}$ and $\mathbf{f}$.

Given an autonomous system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$, define the corresponding flow $\boldsymbol{\phi}_{t}(\mathbf{x})$. What is $\phi_{s}\left(\phi_{t}(\mathbf{x})\right)$ equal to? Define the orbit $\mathcal{O}(\mathbf{x})$ through $\mathbf{x}$ and the limit set $\omega(\mathbf{x})$ of $\mathbf{x}$. Define a homoclinic orbit.