The non-linear evolution of the interchange instability with a poloidal mode number m = 1 and a toroidal mode number n = 1 is investigated in a straight helical system with the ιh= 1 surface in a vacuum magnetic configuration like that of Heliotron E, where ιh, denotes the rotational transform. In the non-linear reduced MHD equations derived by using the stellarator expansion, a thermal-diffusion term and an external-heating term are added to include rapid energy loss and continued heating during the non-linear evolution of the instability. They are solved as an initial and boundary value problem for resistive cylindrical plasmas, in order to explain the sawtooth-like oscillation of electron temperature and density observed in the high-beta current-free Heliotron E experiment. – Without the heating term, the m = 1 pressure-driven mode becomes unstable and changes the pressure profile significantly for central beta, β(0) ≳ 2%, with an initial profile of p(r) ∝ (1 −(r/a)2)2. When a surface-like current increases around the ιh, = 1 resonant surface, because of the diamagnetic effect of the deformed pressure profile, reconnections of the magnetic field lines occur, and the resulting flux surfaces show an m = 2 mode structure. In the case of both thermal diffusion and external heating, the pressure behaviour oscillates between peaked profiles unstable against the m = 1 pressure-driven mode and broad profiles that are stable against it. – These results are consistent with the experimental data.