Fig. 6

Direct numerical simulation of u for a ring solution ($R_1=7$, $R_2=8.629$) perturbed with a linear combination of modes $0,1,2,\dots ,8$. For the given parameters $\beta =0.5$, $\gamma =3$ and Heaviside threshold $h=0.0549$, mode $m=5$ is most unstable, cf. Figure 5. See also the video in Additional file 4.