An existence theorem for invariant manifolds (Q1822395)

For the systems of the form \(\dot x=Ax+f_ 1(x,y,z)\), \(\dot y=By+f_ 2(x,y,z)\), \(\dot z=Cz+f_ 3(x,y,z)\), under the assumption that the spectrum \(\sigma_ A\) is located to the left and the spectrum \(\sigma_ B\) to the right of a vertical straight line \(\ell\) in \({\mathbb{C}}\) the existence of an invariant manifold of the form (x,s(x,z),z) is proved. The two cases (a) \(\sigma_ C\) lies to the left of \(\ell\), (b) \(\sigma_ A\cup \sigma_ C\) cannot be separated from \(\sigma_ B\) by a vertical line in \({\mathbb{C}}\), are considered.