  Any attracting, hyperbolic and proper node of a two-dimensional analytic
vector-field has a unique strong-stable manifold. This manifold is analytic.
The corresponding weak-stable manifolds are, on the other hand, not unique, but
in the nonresonant case there is a unique weak-stable manifold that is
analytic. As the system approaches a saddle-node (under parameter variation), a
sequence of resonances (of increasing order) occur. In this paper, we give a
detailed description of the analytic weak-stable manifolds during this process.
In particular, we relate a ``flapping-mechanism'', corresponding to a dramatic
change of the position of the analytic weak-stable manifold as the parameter
passes through the infinitely many resonances, to the lack of analyticity of
the center manifold at the saddle-node. Our work is motivated and inspired by
the work of Merle, Rapha\"{e}l, Rodnianski, and
  Szeftel, where this flapping mechanism is the crucial ingredient in the
construction of $C^\infty$-smooth self-similar solutions of the compressible
Euler equations.
