Smoothness of class \(C^2\) of nonautonomous linearization without spectral conditions (Q6567212)

The author studies the topological equivalence of a linear system \(\dot x= A(t)\,x\) and a quasilinear system \(\dot y=A(t)\,y+f(t,y)\) on the positive real half line and proves that the smoothness of nonautonomous linearization is of class \(C^2\).\N\NA difference with the paper by \textit{Á. Castañeda} et al. [Proc. R. Soc. Edinb., Sect. A, Math. 150, No. 5, 2484--2502 (2020; Zbl 1467.37026)] is that now the author allows the existence of nonempty unstable manifolds for the linear system which admits a nonuniform dichotomy. For the classic exponential dichotomy the \(C^2\) differentiability is obtained except for a zero Lebesgue set.\N\NThe continuity of the homeomorphism of topological equivalence as a function of two variables (i.e., continuous topological equivalence, as it is called in the aforementioned paper by Á. Castaneda et al.) is not taken into account.