  In this article, a spectral sequence analysis of a filtered Novikov complex
$(\mathcal{N}_{\ast}(f),\Delta)$ over $\mathbb{Z}((t))$ is developed with the
goal of obtaining results relating the algebraic and dynamical settings.
Specifically, the unfolding of a spectral sequence of
$(\mathcal{N}_{\ast}(f),\Delta)$ and the cancellation of its modules is
associated to a one parameter family of circle valued Morse functions on a
surface and the dynamical cancellations of its critical points. The data of a
spectral sequence computed for $(\mathcal{N}_{\ast}(f),\Delta)$ is encoded in a
family of matrices $\Delta^r$ produced by the Spectral Sequence Sweeping
Algorithm (SSSA), which has as its initial input the differential $\Delta$. As
one turns the pages of the spectral sequence, differentials which are
isomorphisms produce cancellation of pairs of modules. Corresponding to these
cancellations, a family of circle-valued Morse functions $f^r$ is obtained by
successively removing the corresponding pairs of critical points of $f$. We
also keep track of all dynamical information on the birth and death of
connecting orbits between consecutive critical points, as well as periodic
orbits that arise within a family of negative gradient flows associated to
$f^r$.

Dynamical Systems;Algebraic Topology