  While it is known that the double-layer Bi$_2$Sr$_2$CaCu$_2$O$_{8+y}$ (BSCCO)
cuprate superconductor exhibits a one-dimensional (1D) incommensurate
superlattice (IS), the effect of IS on the electronic structure remains
elusive. Following the recent shift of the interest from underdoped to optimum
and overdoped phase in BSCCO by increasing the hole doping $x$, controlled by
the oxygen interstitials concentration $y$, here we focus on the multiple
splitting of the density of states (DOS) peaks and emergence of higher order
van Hove singularities (VHS) due to the 1D incommensurate superlattice. It is
known that 1D incommensurate wave vector $\textbf{q} = \epsilon \textbf{b}$
(where $\textbf{b}$ is the reciprocal lattice vector of the orthorhombic
lattice) is controlled by the misfit strain between different atomic layers in
the range $0.209$ - $0.215$ in BSCCO and in the range $0.209$ - $0.25$ in
Bi$_2$Sr$_2$Ca$_{1-x}$Y$_x$Cu$_2$O$_{8+y}$ (BSCYCO). This work reports the
theoretical calculation of a complex pattern of VHS due to the 1D
incommensurate superlattice with large 1D quasi-commensurate supercells with
the wave vector $\epsilon =9/\eta$ in the range $36 > \eta > 43$. The
similarity of the complex VHS splitting and appearing of higher order VHS in a
mismatched CuO$_2$ bilayer with VHS due to the moir{\'e} lattice in strained
twisted bilayer graphene is discussed. This makes mismatched CuO$_2$ bilayer
quite promising for constructing quantum devices with tuned physical
characteristics.
