The formal linearization of a semisimple Lie algebra of vector fields about a singular point

Abstract

A classical theorem by Poincaré gives conditions that a nonlinear ordinary differential equation \[ d x / d t = A ( x ) , dx/dt = A(x), \] with A ( 0 ) = 0 A(0) = 0 in n variables x = ( x 1 , … , x n ) x = ({x_1}, \ldots ,{x_n}) can be reduced to a linear form \[ d x ′ d t = ∂ A ∂ x ( 0 ) x ′ \frac {{dx’}}{{dt}} = \frac {{\partial A}}{{\partial x}}(0)x’ \] by a change of variables x ′ = f ( x ) x’ = f(x) . A generalization is given for a finite set of such differential equations, which form a semisimple Lie algebra.