## African Diaspora Journal of Mathematics

- Afr. Diaspora J. Math. (N.S.)
- Volume 11, Number 1 (2011), 75-102.

### 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.

#### Article information

**Source**

Afr. Diaspora J. Math. (N.S.), Volume 11, Number 1 (2011), 75-102.

**Dates**

First available in Project Euclid: 21 April 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.adjm/1303391947

**Mathematical Reviews number (MathSciNet)**

MR2792212

**Zentralblatt MATH identifier**

1244.93071

**Subjects**

Primary: 37C10: Vector fields, flows, ordinary differential equations

Secondary: 93B17: Transformations 93B18: Linearizations

**Keywords**

Flow-box theorem Frobënius Coordinates transformation PDEs ODEs Lie-derivatives

#### Citation

Tall, Issa Amadou. Flow-Box Theorem and Beyond. Afr. Diaspora J. Math. (N.S.) 11 (2011), no. 1, 75--102. https://projecteuclid.org/euclid.adjm/1303391947