  Hyperbolic umbilic (HU) is a point singularity of the gravitational lens
equation, giving rise to a ring-shaped image formation made of four highly
magnified images, off-centred from the lens centre. Recent observations have
revealed new strongly lensed image formations near HU singularities, and many
more are expected in ongoing and future observations. Like fold/cusp, image
formations near HU also satisfy magnification relation ($R_{\rm hu}$), i.e.,
the signed magnification sum of the four images equals zero. Here, we study how
$R_{\rm hu}$ deviates from zero as a function of area ($A_{\rm hu}$) covered by
the image formation near HU and the distance ($d$) of the central maxima image
(which is part of the HU image formation) from the lens centre for ideal
single- and double-component cluster-scale lenses. For lens ellipticity values
$\geq0.3$, the central maxima image will form sufficiently far from the lens
centre ($d\gtrsim5''$), similar to the observed HU image formations. We also
find that, in some cases, double-component and actual cluster-scale lenses can
lead to large cross-sections for HU(-like) image formations for sources at
$z\gtrsim5$ effectively increasing the chances to observe HU(-like) image
formation at high redshift. Finally, we study the time delay distribution in
the observed HU image formation, finding that not only are these images highly
magnified, but the relative time delay between various pairs of HU
characteristic image formation has a typical value of $\sim100$ days, an order
of magnitude smaller than generic five-image formations in cluster lenses,
making such image formations a promising target for time delay studies.
