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

Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles