Normal forms for periodic orbits of real vector fields (Q943519)

The authors consider the differential system \[ \dot{x}=A\left( t\right) x+f\left( x,t\right)\;,\;\dot{t}=1+g\left( x,t\right), \] where \(A\left( t\right)\in C^{\infty}\left( T_{2\pi},\mathbb{R}^{n\times n}\right)\), \(f\left( x,t\right)\in C^{\infty}\left( U_r\times T_{2\pi},\mathbb{R}^n\right)\) and \(g\left( x,t\right)\in C^{\infty}\left( U_r\times T_{2\pi},\mathbb{R}\right)\) which satisfy \(f\left( x,t\right) = O\left( \| x\| ^2\right)\) and \(g\left( x,t\right) = O\left(\| x\| \right)\) as \(x\to 0\). Note, \(U_r :=\{ x:\| x\| <r\}\subset\mathbb{R}^n\) and \(T_{2\pi}\) is the circle of length \(2\pi\). The main theorem of the paper reads: If the Floquet exponents of the above system are hyperbolic and satisfy non-resonance conditions, then the system is \(C^{\infty}\) locally equivalent to the differential system \[ \dot{y}=A\left( s\right) y+\tilde{f}\left( y,s\right),\quad \dot{s}=1. \]