Typically, the larger kM, the smaller the width of the affected area. This also tells us that a break up in many centers might be hard to see in a computer simulation. Simulations seem to show that the width when kM is very large is actually zero, but here one really has to be careful and study such curves in detail. Most dramatic effects show up when kM is small. Such curves break up into a small number of centers and saddles and affect a large width of curves in phase space.
Tabor shows a qualitative picture of what happens.
If we now increase the perturbation and the centers start to grow radially, we will also get curves around these new centers with rational frequencies that break up. Hence the structure around each center will look similar to the global one with series of new centers bordering the extra centers. But we also have to look at the saddle points. They will give rise to chaos and chaos might set in before the whole scenario described above has fully developed.