  It is shown by numerical simulations within a regularized Biot-Savart law
that dynamical systems of two or three leapfrogging coaxial quantum vortex
rings having a core width $\xi$ and initially placed near a torus of radii
$R_0$ and $r_0$, can be three-dimensionally (quasi-)stable in some regions of
parameters $\Lambda=\ln(R_0/\xi)$ and $W=r_0/R_0$. At fixed $\Lambda$, stable
bands on $W$ are intervals between non-overlapping main parametric resonances
for different (integer) azimuthal wave numbers $m$. The stable intervals are
most wide ($\Delta W\sim$ 0.01--0.05) between $m$-pairs $(1,2)$ and $(2,3)$ at
$\Lambda\approx$ 4--12 thus corresponding to micro/mesoscopic sizes of vortex
rings in the case of superfluid $^4$He. With four and more rings, at least for
$W>0.1$, resonances overlap for all $\Lambda$ and no stable domains exist.
