  We prove several new versions of the Hadamard-Perron Theorem, which relates
infinitesimal dynamics to local dynamics for a sequence of local
diffeomorphisms, and in particular establishes the existence of local stable
and unstable manifolds. Our results imply the classical Hadamard-Perron Theorem
in both its uniform and non-uniform versions, but also apply much more
generally. We introduce a notion of "effective hyperbolicity" and show that if
the rate of effective hyperbolicity is asymptotically positive, then the local
manifolds are well-behaved with positive asymptotic frequency. By applying
effective hyperbolicity to finite orbit segments, we prove a closing lemma
whose conditions can be verified with a finite amount of information.
