Linearization of Vector Fields near Resonant Hyperbolic Rest Points

We study the smoothness of the linearization of a vector field P(ξ)=Aξ+ƒ(ξ) near a resonant hyperbolic fixed point at ξ=0, via a transformation of the form η=ξ+w(ξ), w(0)=0. A is an n×n hyperbolic matrix satisfying an eigenvalue condition Eh of order h≥2, and ƒ(ξ) is of order two near ξ=0. We find a sequence of positive real numbers, h0hq, then there is a Cq linearization in a neighbourhood of the origin. (2) If ƒ(ξ) is only Cp+1, but ƒ(ξ)=R(ξ)+ƒ0(ξ), where R(ξ) is a polynomial of order 2, hqhq, then there is a Cq linearization in a neighbourhood of the origin. (3) If R(ξ) vanishes identically, then Dkw(ξ)≤Cξp−k+1, for some C>0, for k=0, 1,…, q. (4) For planar systems, assume the eigenvalues are (−a, am/n), n>1. Assume that the vector field is Cq+1+r for some qm/n