Nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems (Q403302)

The author develops a theory of the dichotomy (or Sacker-Sell) spectrum for linear ordinary differential equations \(\dot x=A(t)x\) based on nonuniform exponential dichotomies. Here, the dichotomy constants are allowed to grow exponentially in time. Following closely previous work of \textit{S. Siegmund} (see the references [J. Dyn. Differ. Equations 14, No. 1, 243--258 (2002; Zbl 0998.34045); J. Lond. Math. Soc., II. Ser. 65, No. 2, 397--410 (2002; Zbl 1091.34020)]) dealing with uniform dichotomies, a spectral and a decoupling theorem are established.NEWLINENEWLINEThese results constitute the basis to deduce a normal form theory to obtain a smooth simplification of nonlinear nonautonomous ODEs \(\dot x=A(t)x+F(t,x)\), i.e.\ to establish finite jet normal forms. The required non-resonance conditions on the spectral intervals canonically generalize the classical autonomous situation and are those of \textit{S. Siegmund} [J. Differ. Equations 178, No. 2, 541--573 (2002; Zbl 1011.34027)].NEWLINENEWLINEAlthough the concept of nonuniform dichotomies covers a larger class of linear parts, it requires stronger assumptions on the nonlinearities: Indeed, their Taylor-coefficients are required to decay exponentially in time (cf.\ Theorem 1.4). This might limit possible applications, for instance in a nonautonomous bifurcation theory.