The behaviour of the field lines in a torus is analogous to the motion of a non-linear oscillator. If , small and positive, is the perturbation parameter, a toroidal system in which terms of higher order than are assumed to give nonobservable contributions is considered. For large non-linearity (x ≫ 1/2) it was found that two sets of resonances are sufficient to explain the destruction of the magnetic surfaces in the toroidal system. Resonances that transform the unperturbed surfaces into a structure of magnetic islands are called primary resonances, and the secondary resonances transform the bound-statelike contours of a given island into similar structures of secondary magnetic islands. To every primary island two types of stochasticity are attached, an external one due to the overlapping of primary resonances and an internal one due to the overlapping of secondary resonances. Depending on the resonance and the system, the destruction of the magnetic surfaces occurs by either or both processes. For small non-linearity (x ≲ 1/2) the magnetic contours oscillate in a highly irregular fashion and, therefore, overlap causing orbital instabilities. The orbital instabilities are more pronounced for larger fluxes but do not always destroy the flux surfaces at the separatrix. Independently of how small (> 0) is, the flux surfaces are always destroyed at the separatrix, if not by external, then by internal stochasticity. In the immediate neighbourhoods of the elliptic singularities, the field lines are orbitally stable for all non-linearities. These neighbourhoods are small and may become observable for small . For the levitron, the theoretical perturbations for which some of the primary resonances are completely destroyed were calculated. These theoretical values are given in the paper; they are in good agreement with the numerical results.