  In this paper, we investigate a non-interacting scalar field cosmology with
an arbitrary potential using the $f$-deviser method that relies on the
differentiability properties of the potential. Using this alternative
mathematical approach, we present a unified dynamical system analysis at a
scalar field's background and perturbation levels with arbitrary potentials.
For illustration, we consider a monomial and double exponential potential.
These two classes of potentials comprise the asymptotic behaviour of several
classes of scalar field potentials, and, therefore, they provide the skeleton
for the typical behaviour of arbitrary potentials. Moreover, we analyse the
linear cosmological perturbations in the matterless case by considering three
scalar perturbations: the evolution of the Bardeen potentials, the comoving
curvature perturbation, the so-called Sasaki-Mukhanov variable, or the scalar
field perturbation in uniform curvature gauge. Finally, an exhaustive dynamical
system analysis for each scalar perturbation is presented, including the
evolution of Bardeen potentials in the presence of matter.
