In this dissertation, we first develop an alternative method of persistent homology from filtered chain complex level, called Floer-Novikov persistent theory. Then we apply it to study a concrete dynamics problem on quantitatively measuring how far a Hamiltonian flow is from being autonomous. The main results in this thesis are divided into two parts. The first part consists of many results on our Floer-Novikov persistent theory which are analogous to those in classical persistent theory. This includes the important Structure Theorem on the decomposition of a Floer-type complex and Stability Theorem. The main tool we use to develop this theory is non-Archimedean orthogonality and singular value decomposition. The second part consists of the main result that for symplectic manifold in the form of $Sigma_g times M$ (where surface $Sigma_g$ has genus $g geq 4$ and $M$ is {it any} symplectic manifold), the subset of non-autonomous Hamiltonian diffeomorphisms can be arbitrarily far away in Hofer's metric from the group of autonomous Hamiltonian diffeomorphisms. This generalizes Polterovich-Shelukhin's result.