  In this paper we establish the existence of periodic orbits belonging to any
$\sigma$-atoroidal free homotopy class for Hamiltonian systems in the twisted
disc bundle, provided that the compactly supported time-dependent Hamiltonian
function is sufficiently large over the zero section and the magnitude of the
weakly exact $2$-form $\sigma$ admitting a primitive with at most linear growth
on the universal cover is sufficiently small. The proof relies on showing the
invariance of Floer homology under symplectic deformations and on the
computation of Floer homology for the cotangent bundle endowed with its
canonical symplectic form.
  As a consequence, we also prove that, for any nontrivial atoroidal free
homotopy class and any positive finite interval, if the magnitude of a magnetic
field admitting a primitive with at most linear growth on the universal cover
is sufficiently small, the twisted geodesic flow associated to the magnetic
field has a periodic orbit on almost every energy level in the given interval
whose projection to the underlying manifold represents the given free homotopy
class. This application is carried out by showing the finiteness of the
restricted Biran-Polterovich-Salamon capacity.
