Flow-Box Theorem and Beyond

Abstract

For a given vector field $\nu(x)$ around a nonsingular point $x_0$, we provide explicit coordinates $z=\varphi(x)$ in which the vector field is straightened out, i. e., $\varphi_{*}(\nu)(z)=\frac{\partial}{\partial z_1}.$ The procedure is generalized to Frob\"{e}nius Theorem, namely, for an involutive distribution $\Delta={\rm span} \, \left \{\nu_1, \dots, \nu_m \right \}$ around a nonsingular point $x_0$, we give explicit coordinates $z=\varphi(x)$ in which

$$ {\varphi_{*}\Delta= {\rm span} \left \{\frac{\partial}{\partial z_1}, \dots, \frac{\partial}{\partial z_m} \right \}.} $$

The method is illustrated by several examples and is applied to the linearization of control systems.