On Direction of a Vector Field w.r.t. a Given Hyperplane

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 $(0,0)$.

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Hi, please see our FAQ for the scope of this website. In general we ask that questions here be of interest to research mathematicians, and also clearly stated. The question as stated is a bit vague: what is $u(t)$ (is it a given function)? What do you mean by the "direction of a vector field generated on $L$? For given first order system of ODEs, given an arbitrary hyperplane $L$, the flow vector field is generally not going to be tangent to $L$. If you mean the projection of the flow to $L$, in higher dimensions directions can change without going through 0. – Willie Wong Feb 25 2012 at 9:14