A note on the linearization around an equilibrium (Q2277623)

In the paper a dynamical system (1) \(f(y,y',...,y^{(L)})=0\) is studied in the neighbourhood of an equilibrium point \(y=0\) \((f(0,...,0)=0)\) where f: (\({\mathbb{R}}^ q)^{L+1}\to {\mathbb{R}}^ q\) is \(C^{\infty}\). There are given sufficient conditions under which the linearization of (1) is the linear term of the Taylor series.