Stability of the center-stable manifold

Non oscillating solutions of analytic gradient vector fields

“…Let TV^, N 8 , N° be the eigenspaces in TpM of Lp(^) corresponding to the eigenvalues with positive, negative and zero real part, respectively. For any k 6 N there exists a neighbourhood V of P and invariant manifolds for ^ through P in V of class (^w .w^w^w^.w 08 whose tangent spaces at P are respectively TV^, TV 8 , N°^ TV^QTV^, N S (BN C . They are called the unstable, stable, center, center-unstable and centerstable manifold.…”

“…I) We consider first the following situation: the stable manifold W 8 of ^ at P has dimension ^ 2 and | 7 |c W 8 . Then ^ \w« is an hyperbolic vector field (in smaller dimension) with real eigenvalues and 7 is a solution of ^ \w«' This implies that 7 has a well defined tangent as a curve in W 8 and hence as a curve in M. The fact that P' is in the strict transform of W 8 and that there are at most two negative eigenvalues (dim W 8 ^ 2) gives after elementary computation $' € SR^^M'^P') U^^M';?')…”

“…is a consequence of the definition of Go(M; P) in case i = 3 and of the fact that the center manifold is of dimension 1 if i == 2,4. Moreover, P' is in the projectivized tangent space of the formal center manifold of ^ at P. This implies that the linear parts Lp(^) and Lp/(^') have the same eigenvalues (same characteristic polynomials) and thus $ e 5^(M; P) ^ $' e ^^(M'; P') 8 by TT cuts P in a unique point P'. Thus ^(7') = {?'}.…”

“…If they coincide then either F has a single singular point of ^' or F is made of singularities. In any case, for any P' € F such that ^'(P') = 0 we have (8) $' € ^^(M';? '), z = 2 or 4.…”

“…W 8 and $ e ^(M; P). We can suppose that does not have a center manifold consisting of singular points.…”