  Classical conditions for asymptotic stability of periodic solutions
bifurcating from a limit cycle rely on the derivative of the corresponding
bifurcation function F at the bifurcation point t. We show that for analytic
systems this result is the one of topological nature, namely the topological
index ind(t,F) is conclusive versus the sign of the derivative (d/dt)F(t). This
allows to detect asymptotic stability also in the case when (d/dt)F(t)=0.
