Resonance in weak focusing

Since I'm sure you're not yet sick of weak focusing, I want to write a bit about the dangers of resonance in weak focusing systems. In the case I am most familiar with, a storage ring has a (mostly) uniform vertical magnetic field and some strong-focusing quadrupoles to ensure vertical stability. Muons are stored for many hundreds of turns around the ring, so it's critical that the orbits be stable in the 'long' term. The problem is that it's possible to have a localized disruption in the field at just one or two points along the ring. To ensure stability in the long run, the simple betatron oscillations that I mentioned have to be at a different point in their oscillation each time they hit that instability. This has to account for both vertical (frequency $f_y=\sqrt{n}f_C$) and horizontal (frequency $f_x=\sqrt{1-n}f_C$) oscillation frequencies (remember, $f_C$ is the cyclotron frequency, the rate at which the bunches move around the ring). Basically, for the beam to stay stored, any (integer-coefficient) linear combination of these frequencies must not be an integer multiple of the cyclotron frequency. If it is, then every few turns, the beam will be at the same phase of its oscillation at the location of the instability. This is called a resonance, and it can boost the beam out of its stable orbit. As a result, care must be taken in choosing the 'tune' (primarily the magnetic field index) of the storage ring.

This choice can be illustrated with a complicated-looking plot. Below, the x-axis shows $\nu_x=\sqrt{1-n}$, and the y-axis shows $\nu_y=\sqrt{n}$. Integer combinations of $\nu_x$ and $\nu_y$ that yield integers (corresponding to frequencies that are integer multiples of the cyclotron frequency) are plotted as black lines. By the definition of $\nu_x$ and $\nu_y$, we see that $\nu_x^2+\nu_y^2=1$, which is shown on the plot as a red curve. Blue dots are along the intersection of this tune curve and the forbidden lines, and represent bad tunes, while the bright green dots show acceptable tunes, at least to the degree that we've plotted the resonance lines. Neah, eh?