  We investigate the ground state magnetic configurations of a Fibonacci chain
of classical spins with nearest-neighbor ferromagnetic and monoaxial
Dzyaloshinskii-Moriya exchange interactions. Our analysis reveals a diverse
array of magnetic textures induced by an external magnetic field perpendicular
to the Dzyaloshinskii-Moriya vector. These textures exhibit a spectrum ranging
from a quasi-fully polarized non-collinear state under high magnetic fields,
capable of maintaining metastable chiral soliton topological defects, to a
variety of quasicrystalline chiral soliton lattices below a critical field
$H_c$. For a range of magnetic fields below $H_c$, the ground state spin
textures result from the interplay between an effective quasiperiodic potential
influencing the solitons and their repulsive interactions. At lower magnetic
fields, the system experiences a commensurate-incommensurate transition,
signified by the appearance of discommensurations in the quasicrystalline
soliton lattice. In the absence of an external magnetic field, the ground state
assumes a helical configuration with a quasiperiodic pitch angle.
