nodal lines show wave dislocations typical of non-separable wave equations. Non-separability goes along with the chaotic ray dynamics of Fig. \[fig:Poincare-surface-of-section\]. Despite the increasingly complex nodal structure, all modes share the whispering-gallery like wave propagation along the convex parts of the boundary.

![Field intensities for the modes corresponding to the first four Q-factor peaks in Fig. \[fig:Wavenumber-sweeps\] for $n=2.4$, $\epsilon=0.444$. The Q factors are (a) $2500$, (b) $2100$, (c) $2400$ and (d) $4500$. \[fig:Field-intensities-forPeaks\]](Figure7){width="0.6\columnwidth"}

The Q-factor of any given mode may depend non-monotonically on the deformation parameter $\epsilon$ of Eq. (\[eq:ShapeCurve\]), as shown in Fig. \[fig:Q-factor-versusEps\]. However, we find other high-Q modes over the entire range of deformations in Fig. \[fig:Q-factor-versusEps\], as illustrated by the examples in Fig. \[fig:Field-intensitiesVaryEps\]. The robustness of the folded morphology against deformation is another interesting feature that these modes share with conventional whispering-gallery modes in convex resonators.

![Field intensities for folded WGMs at different deformations: (a) $\epsilon=0.52$, (b) $\epsilon=0.6$, (c) $\epsilon=0.7$. The refractive index is $n=2.4$. In (c), a smaller wavenumber is chosen, whereas (a) and (b) have comparable wavenumbers. The narrowing of the horizontal stubs is a result of the requirement that its tangents must match the curve described by Eq. (\[eq:ShapeCurve\]) at the 