  Consider a nonlinear Schr\"odinger equation in $R^3$ whose linear part has
three or more eigenvalues satisfying some resonance conditions. Solutions which
are initially small in $H^1 \cap L^1(R^3)$ and inside a neighborhood of the
first excited state family are shown to converge to either a first excited
state or a ground state at time infinity. An essential part of our analysis is
on the linear and nonlinear estimates near nonlinear excited states, around
which the linearized operators have eigenvalues with nonzero real parts and
their corresponding eigenfunctions are not uniformly localized in space.
