Flows about a critical point with a single zero characteristic root

Abstract

where x is a real (B + I)-vector. Here A is a real, constant, (n + 1) x (n -+ I) matrix with one zero characteristic root and all remaining characteristic roots have real parts of the same sign, X(x) is a real analytic vector function starting with terms of order greater than one, and x = 0 is an isolated critical point. Such systems have been studied by Bendixson, Liapunov, Mendo one consequence of his work is that (i) has a unique analytic stable (unstable) manifold when the nonzero characteristic roots of A all have negative (positive) real parts. That thii stable (u&table) manifold is the only c1 stable (unstable) manifold is implied by the wok of Hartman [3, pp. 2342431, A more recent concept than the stable manifold is’ that of a c&ter manifold for a system which has ehar~c~erist~~ roots itith zero real parts. The existence of a C” m~old for (i), for :any positive integer m, is an