If there is a very small perturbation δf to the field f , then how about the change in the magnetization? The answer is the occurrence of fractal structure in the magnetic de Haas- van Alphen (dHvA) oscillations, which is very important for the experimental measurement of the Fermi surface topology. Now we investigate this fractal structure in the magnetic oscillations for the present system. For convenience and simplicity, we choose the field f =1 / 4 and the perturbation δf = − 1 / 196 ( f ′ = f + δf =12 / 49). The calculated magnetization M at the fields f and f ′ are plotted in Fig. 6. From Fig. 6 one can observe the following features: (i) In the insulating region, the magnitudes of the magnetization M have little difference at the fields f and f ′ . The perturbation δf is more small, more little in the changes of the magnetization M ; (ii) In the metallic regions at the field f , however, there are dHvA oscillations appearing in the magnetization M at the field f + δf . The perturbation δf is more small, the magnetization oscillates more rapidly; (iii) The collapsed middle band originally at field f =1 / 4 now spreads at f ′ =12 / 49. And in this middle band there are also dHvA oscillations appearing in the orbital magnetization. To see this fractal feature more clearly, we enlarge the magnetization in the inset in Fig. 6; (iv) There are similar variations in the two components of M (see Fig. 7). In the insulating regions at f =1 / 4, there are little difference in the conventional and Berry-phase terms. However, both terms exhibit different behaviors in the metallic regions. While the Berry-phase term has little difference at different fields as it in the insulating regions, there are many quantum steps appearing in the conventional term at f ′ =12 / 49. The reason for these quantum steps appearing in the conventional magnetization is that the topology of the energy spectrum at the field f ′ = f + δf is different from that at the field f . When the external field f changes to f + δf , the eigenbands at the field f split into lots of subbands. The perturbation δf is more small, the number of the splitting subbands is more large. Because the upper limit of the k -integral of m$_{n}$ ( k ) is invariant as the Fermi level µ$_{0}$ varies in the subgap, then the conventional term in the subgap keeps a constant, which is the reason for the quantum step appearing in the conventional term. On the other hand, the integral of the Berry-phase term intimately depends on the Fermi level and has no such relation with the energy gaps (or bands). That results in the little difference in the Berry- phase term at different fields. So, when one changes the electron’s fillings through these splitting bands, the total magnetization then exhibits the fractal structure in the magnetic dHvA oscillations. 11