Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S19 / 1 / b / Solution

For a periodic uncoupled lattice, $\mathrm{tr}M=2\cos \mu$ in the Twiss representation. Here $\cos \mu =1-L_1{L}_2{q}^2/2$, so bounded nondegenerate cell motion requiresThe zero-strength endpoint is only marginal and usually gives a drifting Jordan block. The phase advance increases from zero to $\pi$ on the positive-strength stable branch. Repeating the cell $N$ times gives ring eigenvalues $e^{\pm iN\mu }$ and tune $N\mu /(2\pi )$; it does not enlarge the stable strength interval. Integer ring phase advances can be resonance-sensitive in the presence of errors even when the ideal repeated matrix is bounded.