  The ground state of large Hubbard $U$ limit of a honeycomb lattice near
half-filling is known to be a singlet $d+id$-wave superconductor. It is also
known that this $d+id$ superconductor exhibits a chiral $p+ip$ pairing locally
at the Dirac cone, characterized by a $2\mathbb{Z}$ topological invariant. By
constructing a dual transformation, we demonstrate that this $2\mathbb{Z}$
topological superconductor is equivalent to a collection of two topological
ferromagnetic insulators. As a result of the duality, the topology of the
electronic structures for a $d+id$ superconductor is controllable via the
change of the chemical potential by tuning the gate voltage. In particular,
instead of being always a chiral superconductor, we find that the $d+id$
superconductor undergoes a topological phase transition from a chiral
superconductor to a quasi-helical superconductor as the gap amplitude or the
chemical potential decreases. The quasi-helical superconducting phase is found
to be characterized by a topological invariant in the pseudo-spin charge sector
with vanishing both the Chern number and the spin Chern number. We further
elucidate the topological phase transition by analyzing the relationship
between the topological invariant and the rotation symmetry. Due to the angular
momentum carried by the gap function and spin-orbit interactions, we show that
by placing $d+id$ superconductors in proximity to ferromagnets, varieties of
chiral superconducting phases characterized by higher Chern numbers can be
accessed, providing a new platform for hosting large numbers of Majorana modes
at edges.
