  We study numerically the dynamical system of a two-electron atom with the
Darwin interaction as a model to investigate scale-dependent effects of the
relativistic action-at-a-distance electrodynamics. This dynamical system
consists of a small perturbation of the Coulomb dynamics for energies in the
atomic range. The key properties of the Coulomb dynamics are: (i) a peculiar
mixed-type phase space with sparse families of stable non-ionizing orbits and
(ii) scale-invariance symmetry, with all orbits defined by an arbitrary scale
parameter. The combination of this peculiar chaotic dynamics ((i) and (ii)),
with the scale-dependent relativistic corrections (Darwin interaction)
generates the phenomenon of scale-dependent stability: We find numerical
evidence that stable non-ionizing orbits can exist only for a discrete set of
resonant energies. The Fourier transform of these non-ionizing orbits is a set
of sharp frequencies. The energies and sharp frequencies of the non-ionizing
orbits we study are in the quantum atomic range.
