  In this paper, we develop an approach to the problem of closing lemma based
on KAM normal form. The new approach differs from existing $C^1$ perturbation
approach and spectral approach, and can handle the high regularity, high
dimensional cases and even Riemannian metric perturbations. Moreover, the proof
is constructive and effective. We apply the method to the original nearly
integrable setting of Poincar\'e and confirm several old and new conjectures
with weak formulations. First, for Poincar\'e's original setting of nearly
integrable systems, we prove that for typical perturbations, periodic orbits
are asymptotically dense as the size of perturbation tends to zero. Second, we
prove that typical smooth perturbation of the geodesic flow on the flat torus
has asymptotically dense periodic orbits, which partially solves an open
problem since Pugh-Robinson's $C^1$-closing lemma. Third, we prove that for
typical Hamiltonian or contact perturbation of the geodesic flows of the
ellipsoid has asymptotically dense orbit on the energy level, which enhances
the recent researches on strong closing lemma, and also confirms partially a
conjecture of Fish-Hofer in this setting. We also discuss the relation of our
models to the recent researches on many-body localization in physics.
