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Flow-Box Theorem and Beyond

Volume 11, Number 1 (2011), 75 - 102

Flow-Box Theorem and Beyond

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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 Frobënius Theorem, namely, for an involutive distribution $\Delta = span \{\nu_1, ..., \nu_m \}$ 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.