I'm learning continuous flows and I found this definition:
Let $(X,d)$ be a compact metric space and $\phi:\mathbb{R}\times X\rightarrow X$ be a flow continuous. Denote by $\mathcal{H}$ the set of continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. Given $x\in X$ and $\delta>0$ Bowen walters define the dynamical ball: $$ \Gamma^{\phi}_{\delta}(x)=\bigcup_{h\in\mathcal{H}}\bigcap_{t\in \mathbb{R}}\phi_{-h(t)}(B[\phi_t(x),\delta ]). $$ i.e. $z\in \Gamma^{\phi}_{\delta}(x)$ iff there exist $h\in\mathcal{H}$ such that $d(\phi(t,x),\phi(h(t),z))\leq \delta$ for all $t\in \mathbb{R}$.
Trying to understand this ball, for example by taking the flow generated by the equation $(x^{\prime}, y^{\prime})=(-y,x)$ in the plane it holds that $\Gamma^{\phi}_{\delta}(x)=B[x,\delta ]$. I wonder then if for all flow in $X$ this dynamical ball it will be a closed set of $X$? or at least this ball it will be a Borel set of $X$?
Strictly speaking Bowen and Walters never defined the ball that you mention. In their only joint work, they do consider related problems, but the idea is instead to ask whether points in $\Gamma^\phi_\delta(x)$, for $\delta$ sufficiently small, are in the orbit of $x$.
Anyways, $\Gamma^\phi_\delta(x)$ is always a Borel set provided that $\phi$ is continuous on $\mathbb R\times X$. Sometimes it is closed, sometimes not. The interesting examples to consider are not isometries, and so the differential equation that you chose is precisely the main type that produces no nontrivial results (or if you prefer: flows acting by isometries are not expansive).