Poincaré type theorems for non-autonomous systems
The authors develop a theory of normal forms for the case of nonautonomous systems of differential equations. Consider an analytic system of differential equations \[ x'=F(x,t)=A(t)x+f(x,t), \tag{1} \] where \(f=O(\|x\|^2)\), and its linearization, \[ x'=A(t)x. \] In terms of the Sacker-Sell spectrum of the linearization, they give conditions under which (i) any system \[ x'=G(x,t) \] with \(G-f=O(\|x\|^d)\) is locally analytically equivalent to (1) (equivalence in the jet class); (ii) system (1) is locally analytically equivalent to its linearization (Poincaré type); (iii) system (1) is locally analytically equivalent to a system \[ x'=A(t)x+g(x,t), \] where \(g\) is a polynomial in \(x\) (Poincaré-Dulac type).
scientific article; zbMATH DE number 5365158
scientific article