A nonlinear system modeled as a set of ODE equations as $f( {\bf x}(t), {\bf \dot{x}}(t), {\bf u}(t))=0$. Let $L$ be a hyper-plane in $\mathbb{R}^n$, described by $L= \{ {\bf x} | A {\bf x} = b \}$.

What is the direction of the vector field generated by the $f$ on $L$? Preferred answer should be in a form of a function $g({\bf x})$ where the output of g is positive in one direction and negative in the other case.

I appreciate any hint or directions toward the answer.

For example, in $\mathbb{R}^2$, for a system described by

$x' = -2x + 2y $

$y' = -2x - 2y$

and for a line $y=x$, the direction of the vector field changes at point $(0,0)$.