Dynamical system and level curves Is it possible to design a function $f(x, z) = 0$ from a dynamical system $\dot{x} = a(x)$ such that the trace of trajectories are the level curves of $f$, i.e. $f(x, c) = 0$, $c \in \mathbb{R}$?
 A: Locally, I believe this is possible as an application of Frobenius' theorem. Globally this is false. To see why, consider the linear system
$$
\dot{x}(t) = -x(t).
$$
Solutions to this differential equation, $x(t) = A e^{-t},$ cannot be described globally as level sets in terms of $x$ alone. This follows from the fact that the image of the solution in the state-space $\mathbb{R}$ is not a closed set for $A\neq 0$. As a result, no continuous function $f$ exists that renders that class of solutions to the DE as a level set.
As another example, consider $x \in \mathbb{R}^2$ with dynamics
$$
\dot{x}(t) = 1 - (x_1^2 + x_2^2).
$$
The trajectories that tend towards the unit circle cannot be treated as level sets globally. Worse still, even the invariant points on the unit circle cannot be treated as level sets of a continuous function $f: S^1 \to U \subseteq\mathbb{R}$  unless you relax one-to-oneness (one number in the codomain corresponds to exactly one trajectory in the domain) or have $U$ be some "half-open" set.
