  We study mathematical structures on the moduli spaces of BPS structures of
$\mathcal{N}=2$ theories. Guided by the realization of BPS structures within
type IIB string theory on non-compact Calabi-Yau threefolds, we develop a
notion of BPS variation of Hodge structure which gives rise to special K\"ahler
geometry as well as to Picard-Fuchs equations governing the central charges of
the BPS structure. We focus our study on cases with complex one dimensional
moduli spaces and charge lattices of rank two including Argyres-Douglas $A_2$
as well as Seiberg-Witten $SU(2)$ theories. In these cases the moduli spaces
are identified with modular curves and we determine the expressions of the
central charges in terms of quasi-modular forms of the corresponding duality
groups. We furthermore determine the curves of marginal stability and study the
attractor flow in these examples, showing that it provides another way of
determining the complete BPS spectrum in these cases.
