Abstract The fluxon dynamic in the Josephson chains with magnetic inclusions is investigated. In the framework of the McLathlin and Scott perturbation theory it has been shown that fluxons and antifluxons have different interaction energies with magnetic inclusions. The fluxons may be in the bound state with the magnetic inclusion and there is a distance between the centre of fluxon and the magnetic inclusion. The critical current (through the junction) at which the fluxon unbinding takes place is determined. The frequency of the small oscillations of the fluxon in the equilibrium state is calculated. It is shown that there are conditions for the oscillation mode softening. A new type of electronic device is discussed.